René Vidal

Liangzu Peng, Manolis C. Tsakiris, René Vidal

This paper is about the old Wahba problem in its more general form, which we call "simultaneous rotation and correspondence search". In this generalization we need to find a rotation that best aligns two partially overlapping $3$D point sets, of sizes $m$ and $n$ respectively with $m\geq n$. We first propose a solver, $\texttt{ARCS}$, that i) assumes noiseless point sets in general position, ii) requires only $2$ inliers, iii) uses $O(m\log m)$ time and $O(m)$ space, and iv) can successfully solve the problem even with, e.g., $m,n\approx 10^6$ in about $0.1$ seconds. We next robustify $\texttt{ARCS}$ to noise, for which we approximately solve consensus maximization problems using ideas from robust subspace learning and interval stabbing. Thirdly, we refine the approximately found consensus set by a Riemannian subgradient descent approach over the space of unit quaternions, which we show converges globally to an $\varepsilon$-stationary point in $O(\varepsilon^{-4})$ iterations, or locally to the ground-truth at a linear rate in the absence of noise. We combine these algorithms into $\texttt{ARCS+}$, to simultaneously search for rotations and correspondences. Experiments show that $\texttt{ARCS+}$ achieves state-of-the-art performance on large-scale datasets with more than $10^6$ points with a $10^4$ time-speedup over alternative methods. \url{https://github.com/liangzu/ARCS}

Tianzhe Chu, Shengbang Tong, Tianjiao Ding et al.

The advent of large pre-trained models has brought about a paradigm shift in both visual representation learning and natural language processing. However, clustering unlabeled images, as a fundamental and classic machine learning problem, still lacks an effective solution, particularly for large-scale datasets. In this paper, we propose a novel image clustering pipeline that leverages the powerful feature representation of large pre-trained models such as CLIP and cluster images effectively and efficiently at scale. We first developed a novel algorithm to estimate the number of clusters in a given dataset. We then show that the pre-trained features are significantly more structured by further optimizing the rate reduction objective. The resulting features may significantly improve the clustering accuracy, e.g., from 57\% to 66\% on ImageNet-1k. Furthermore, by leveraging CLIP's multimodality bridge between image and text, we develop a simple yet effective self-labeling algorithm that produces meaningful captions for the clusters. Through extensive experiments, we show that our pipeline works well on standard datasets such as CIFAR-10, CIFAR-100, and ImageNet-1k. It also extends to datasets that are not curated for clustering, such as LAION-Aesthetics and WikiArts. We released the code in https://github.com/LeslieTrue/CPP.

Kaleab A. Kinfu, René Vidal

Adversarial training (AT) is a simple yet effective defense against adversarial attacks to image classification systems, which is based on augmenting the training set with attacks that maximize the loss. However, the effectiveness of AT as a defense for video classification has not been thoroughly studied. Our first contribution is to show that generating optimal attacks for video requires carefully tuning the attack parameters, especially the step size. Notably, we show that the optimal step size varies linearly with the attack budget. Our second contribution is to show that using a smaller (sub-optimal) attack budget at training time leads to a more robust performance at test time. Based on these findings, we propose three defenses against attacks with variable attack budgets. The first one, Adaptive AT, is a technique where the attack budget is drawn from a distribution that is adapted as training iterations proceed. The second, Curriculum AT, is a technique where the attack budget is increased as training iterations proceed. The third, Generative AT, further couples AT with a denoising generative adversarial network to boost robust performance. Experiments on the UCF101 dataset demonstrate that the proposed methods improve adversarial robustness against multiple attack types.

Darshan Thaker, Paris Giampouras, René Vidal

Deep neural network-based classifiers have been shown to be vulnerable to imperceptible perturbations to their input, such as $\ell_p$-bounded norm adversarial attacks. This has motivated the development of many defense methods, which are then broken by new attacks, and so on. This paper focuses on a different but related problem of reverse engineering adversarial attacks. Specifically, given an attacked signal, we study conditions under which one can determine the type of attack ($\ell_1$, $\ell_2$ or $\ell_\infty$) and recover the clean signal. We pose this problem as a block-sparse recovery problem, where both the signal and the attack are assumed to lie in a union of subspaces that includes one subspace per class and one subspace per attack type. We derive geometric conditions on the subspaces under which any attacked signal can be decomposed as the sum of a clean signal plus an attack. In addition, by determining the subspaces that contain the signal and the attack, we can also classify the signal and determine the attack type. Experiments on digit and face classification demonstrate the effectiveness of the proposed approach.

Kaleab A. Kinfu, Rene Vidal

While Convolutional Neural Networks (CNNs) have been widely successful in 2D human pose estimation, Vision Transformers (ViTs) have emerged as a promising alternative to CNNs, boosting state-of-the-art performance. However, the quadratic computational complexity of ViTs has limited their applicability for processing high-resolution images. In this paper, we propose three methods for reducing ViT's computational complexity, which are based on selecting and processing a small number of most informative patches while disregarding others. The first two methods leverage a lightweight pose estimation network to guide the patch selection process, while the third method utilizes a set of learnable joint tokens to ensure that the selected patches contain the most important information about body joints. Experiments across six benchmarks show that our proposed methods achieve a significant reduction in computational complexity, ranging from 30% to 44%, with only a minimal drop in accuracy between 0% and 3.5%.

Christiaan Lamers, Rene Vidal, Nabil Belbachir et al.

We consider the problem of learning multiple tasks in a continual learning setting in which data from different tasks is presented to the learner in a streaming fashion. A key challenge in this setting is the so-called "catastrophic forgetting problem", in which the performance of the learner in an "old task" decreases when subsequently trained on a "new task". Existing continual learning methods, such as Averaged Gradient Episodic Memory (A-GEM) and Orthogonal Gradient Descent (OGD), address catastrophic forgetting by minimizing the loss for the current task without increasing the loss for previous tasks. However, these methods assume the learner knows when the task changes, which is unrealistic in practice. In this paper, we alleviate the need to provide the algorithm with information about task changes by using an online clustering-based approach on a dynamically updated finite pool of samples or gradients. We thereby successfully counteract catastrophic forgetting in one of the hardest settings, namely: domain-incremental learning, a setting for which the problem was previously unsolved. We showcase the benefits of our approach by applying these ideas to projection-based methods, such as A-GEM and OGD, which lead to task-agnostic versions of them. Experiments on real datasets demonstrate the effectiveness of the proposed strategy and its promising performance compared to state-of-the-art methods.

Aditya Chattopadhyay, Stewart Slocum, Benjamin D. Haeffele et al.

There is a growing concern about typically opaque decision-making with high-performance machine learning algorithms. Providing an explanation of the reasoning process in domain-specific terms can be crucial for adoption in risk-sensitive domains such as healthcare. We argue that machine learning algorithms should be interpretable by design and that the language in which these interpretations are expressed should be domain- and task-dependent. Consequently, we base our model's prediction on a family of user-defined and task-specific binary functions of the data, each having a clear interpretation to the end-user. We then minimize the expected number of queries needed for accurate prediction on any given input. As the solution is generally intractable, following prior work, we choose the queries sequentially based on information gain. However, in contrast to previous work, we need not assume the queries are conditionally independent. Instead, we leverage a stochastic generative model (VAE) and an MCMC algorithm (Unadjusted Langevin) to select the most informative query about the input based on previous query-answers. This enables the online determination of a query chain of whatever depth is required to resolve prediction ambiguities. Finally, experiments on vision and NLP tasks demonstrate the efficacy of our approach and its superiority over post-hoc explanations.

Danny Weyns, Thomas Baeck, Rene Vidal et al.

Computing systems are omnipresent; their sustainability has become crucial for our society. A key aspect of this sustainability is the ability of computing systems to cope with the continuous change they face, ranging from dynamic operating conditions, to changing goals, and technological progress. While we are able to engineer smart computing systems that autonomously deal with various types of changes, handling unanticipated changes requires system evolution, which remains in essence a human-centered process. This will eventually become unmanageable. To break through the status quo, we put forward an arguable opinion for the vision of self-evolving computing systems that are equipped with an evolutionary engine enabling them to evolve autonomously. Specifically, when a self-evolving computing system detects conditions outside its operational domain, such as an anomaly or a new goal, it activates an evolutionary engine that runs online experiments to determine how the system needs to evolve to deal with the changes, thereby evolving its architecture. During this process the engine can integrate new computing elements that are provided by computing warehouses. These computing elements provide specifications and procedures enabling their automatic integration. We motivate the need for self-evolving computing systems in light of the state of the art, outline a conceptual architecture of self-evolving computing systems, and illustrate the architecture for a future smart city mobility system that needs to evolve continuously with changing conditions. To conclude, we highlight key research challenges to realize the vision of self-evolving computing systems.

Aditya Chattopadhyay, Kwan Ho Ryan Chan, Benjamin D. Haeffele et al.

There is a growing interest in the machine learning community in developing predictive algorithms that are "interpretable by design". Towards this end, recent work proposes to make interpretable decisions by sequentially asking interpretable queries about data until a prediction can be made with high confidence based on the answers obtained (the history). To promote short query-answer chains, a greedy procedure called Information Pursuit (IP) is used, which adaptively chooses queries in order of information gain. Generative models are employed to learn the distribution of query-answers and labels, which is in turn used to estimate the most informative query. However, learning and inference with a full generative model of the data is often intractable for complex tasks. In this work, we propose Variational Information Pursuit (V-IP), a variational characterization of IP which bypasses the need for learning generative models. V-IP is based on finding a query selection strategy and a classifier that minimizes the expected cross-entropy between true and predicted labels. We then demonstrate that the IP strategy is the optimal solution to this problem. Therefore, instead of learning generative models, we can use our optimal strategy to directly pick the most informative query given any history. We then develop a practical algorithm by defining a finite-dimensional parameterization of our strategy and classifier using deep networks and train them end-to-end using our objective. Empirically, V-IP is 10-100x faster than IP on different Vision and NLP tasks with competitive performance. Moreover, V-IP finds much shorter query chains when compared to reinforcement learning which is typically used in sequential-decision-making problems. Finally, we demonstrate the utility of V-IP on challenging tasks like medical diagnosis where the performance is far superior to the generative modelling approach.

Yashar Deldjoo, Zhankui He, Julian McAuley et al.

Generative models are a class of AI models capable of creating new instances of data by learning and sampling from their statistical distributions. In recent years, these models have gained prominence in machine learning due to the development of approaches such as generative adversarial networks (GANs), variational autoencoders (VAEs), and transformer-based architectures such as GPT. These models have applications across various domains, such as image generation, text synthesis, and music composition. In recommender systems, generative models, referred to as Gen-RecSys, improve the accuracy and diversity of recommendations by generating structured outputs, text-based interactions, and multimedia content. By leveraging these capabilities, Gen-RecSys can produce more personalized, engaging, and dynamic user experiences, expanding the role of AI in eCommerce, media, and beyond. Our book goes beyond existing literature by offering a comprehensive understanding of generative models and their applications, with a special focus on deep generative models (DGMs) and their classification. We introduce a taxonomy that categorizes DGMs into three types: ID-driven models, large language models (LLMs), and multimodal models. Each category addresses unique technical and architectural advancements within its respective research area. This taxonomy allows researchers to easily navigate developments in Gen-RecSys across domains such as conversational AI and multimodal content generation. Additionally, we examine the impact and potential risks of generative models, emphasizing the importance of robust evaluation frameworks.

Aditya Chattopadhyay, Xi Zhang, David Paul Wipf et al.

We present a structured graph variational autoencoder for generating the layout of indoor 3D scenes. Given the room type (e.g., living room or library) and the room layout (e.g., room elements such as floor and walls), our architecture generates a collection of objects (e.g., furniture items such as sofa, table and chairs) that is consistent with the room type and layout. This is a challenging problem because the generated scene should satisfy multiple constrains, e.g., each object must lie inside the room and two objects cannot occupy the same volume. To address these challenges, we propose a deep generative model that encodes these relationships as soft constraints on an attributed graph (e.g., the nodes capture attributes of room and furniture elements, such as class, pose and size, and the edges capture geometric relationships such as relative orientation). The architecture consists of a graph encoder that maps the input graph to a structured latent space, and a graph decoder that generates a furniture graph, given a latent code and the room graph. The latent space is modeled with auto-regressive priors, which facilitates the generation of highly structured scenes. We also propose an efficient training procedure that combines matching and constrained learning. Experiments on the 3D-FRONT dataset show that our method produces scenes that are diverse and are adapted to the room layout.

Hancheng Min, Enrique Mallada, René Vidal

This paper studies the problem of training a two-layer ReLU network for binary classification using gradient flow with small initialization. We consider a training dataset with well-separated input vectors: Any pair of input data with the same label are positively correlated, and any pair with different labels are negatively correlated. Our analysis shows that, during the early phase of training, neurons in the first layer try to align with either the positive data or the negative data, depending on its corresponding weight on the second layer. A careful analysis of the neurons' directional dynamics allows us to provide an $\mathcal{O}(\frac{\log n}{\sqrtμ})$ upper bound on the time it takes for all neurons to achieve good alignment with the input data, where $n$ is the number of data points and $μ$ measures how well the data are separated. After the early alignment phase, the loss converges to zero at a $\mathcal{O}(\frac{1}{t})$ rate, and the weight matrix on the first layer is approximately low-rank. Numerical experiments on the MNIST dataset illustrate our theoretical findings.

Yutao Tang, Benjamín Béjar, Joey K. -Y. Essoe et al.

Tourette Syndrome (TS) is a behavior disorder that onsets in childhood and is characterized by the expression of involuntary movements and sounds commonly referred to as tics. Behavioral therapy is the first-line treatment for patients with TS, and it helps patients raise awareness about tic occurrence as well as develop tic inhibition strategies. However, the limited availability of therapists and the difficulties for in-home follow up work limits its effectiveness. An automatic tic detection system that is easy to deploy could alleviate the difficulties of home-therapy by providing feedback to the patients while exercising tic awareness. In this work, we propose a novel architecture (T-Net) for automatic tic detection and classification from untrimmed videos. T-Net combines temporal detection and segmentation and operates on features that are interpretable to a clinician. We compare T-Net to several state-of-the-art systems working on deep features extracted from the raw videos and T-Net achieves comparable performance in terms of average precision while relying on interpretable features needed in clinical practice.

Liangzu Peng, Paris V. Giampouras, René Vidal

The goal of continual learning is to find a model that solves multiple learning tasks which are presented sequentially to the learner. A key challenge in this setting is that the learner may forget how to solve a previous task when learning a new task, a phenomenon known as catastrophic forgetting. To address this challenge, many practical methods have been proposed, including memory-based, regularization-based, and expansion-based methods. However, a rigorous theoretical understanding of these methods remains elusive. This paper aims to bridge this gap between theory and practice by proposing a new continual learning framework called Ideal Continual Learner (ICL), which is guaranteed to avoid catastrophic forgetting by construction. We show that ICL unifies multiple well-established continual learning methods and gives new theoretical insights into the strengths and weaknesses of these methods. We also derive generalization bounds for ICL which allow us to theoretically quantify how rehearsal affects generalization. Finally, we connect ICL to several classic subjects and research topics of modern interest, which allows us to make historical remarks and inspire future directions.

Ambar Pal, Jeremias Sulam, René Vidal

The susceptibility of modern machine learning classifiers to adversarial examples has motivated theoretical results suggesting that these might be unavoidable. However, these results can be too general to be applicable to natural data distributions. Indeed, humans are quite robust for tasks involving vision. This apparent conflict motivates a deeper dive into the question: Are adversarial examples truly unavoidable? In this work, we theoretically demonstrate that a key property of the data distribution -- concentration on small-volume subsets of the input space -- determines whether a robust classifier exists. We further demonstrate that, for a data distribution concentrated on a union of low-dimensional linear subspaces, utilizing structure in data naturally leads to classifiers that enjoy data-dependent polyhedral robustness guarantees, improving upon methods for provable certification in certain regimes.

Juan Cervino, Luiz F. O. Chamon, Benjamin D. Haeffele et al.

Smoothness and low dimensional structures play central roles in improving generalization and stability in learning and statistics. This work combines techniques from semi-infinite constrained learning and manifold regularization to learn representations that are globally smooth on a manifold. To do so, it shows that under typical conditions the problem of learning a Lipschitz continuous function on a manifold is equivalent to a dynamically weighted manifold regularization problem. This observation leads to a practical algorithm based on a weighted Laplacian penalty whose weights are adapted using stochastic gradient techniques. It is shown that under mild conditions, this method estimates the Lipschitz constant of the solution, learning a globally smooth solution as a byproduct. Experiments on real world data illustrate the advantages of the proposed method relative to existing alternatives.

Liangzu Peng, Mahyar Fazlyab, René Vidal

The rotation search problem aims to find a 3D rotation that best aligns a given number of point pairs. To induce robustness against outliers for rotation search, prior work considers truncated least-squares (TLS), which is a non-convex optimization problem, and its semidefinite relaxation (SDR) as a tractable alternative. Whether this SDR is theoretically tight in the presence of noise, outliers, or both has remained largely unexplored. We derive conditions that characterize the tightness of this SDR, showing that the tightness depends on the noise level, the truncation parameters of TLS, and the outlier distribution (random or clustered). In particular, we give a short proof for the tightness in the noiseless and outlier-free case, as opposed to the lengthy analysis of prior work.

Anton Korikov, Scott Sanner, Yashar Deldjoo et al.

While previous chapters focused on recommendation systems (RSs) based on standardized, non-verbal user feedback such as purchases, views, and clicks -- the advent of LLMs has unlocked the use of natural language (NL) interactions for recommendation. This chapter discusses how LLMs' abilities for general NL reasoning present novel opportunities to build highly personalized RSs -- which can effectively connect nuanced and diverse user preferences to items, potentially via interactive dialogues. To begin this discussion, we first present a taxonomy of the key data sources for language-driven recommendation, covering item descriptions, user-system interactions, and user profiles. We then proceed to fundamental techniques for LLM recommendation, reviewing the use of encoder-only and autoregressive LLM recommendation in both tuned and untuned settings. Afterwards, we move to multi-module recommendation architectures in which LLMs interact with components such as retrievers and RSs in multi-stage pipelines. This brings us to architectures for conversational recommender systems (CRSs), in which LLMs facilitate multi-turn dialogues where each turn presents an opportunity not only to make recommendations, but also to engage with the user in interactive preference elicitation, critiquing, and question-answering.

Yutao Tang, Benjamin Bejar, Rene Vidal

Recent work on action recognition leverages 3D features and textual information to achieve state-of-the-art performance. However, most of the current few-shot action recognition methods still rely on 2D frame-level representations, often require additional components to model temporal relations, and employ complex distance functions to achieve accurate alignment of these representations. In addition, existing methods struggle to effectively integrate textual semantics, some resorting to concatenation or addition of textual and visual features, and some using text merely as an additional supervision without truly achieving feature fusion and information transfer from different modalities. In this work, we propose a simple yet effective Semantic-Aware Few-Shot Action Recognition (SAFSAR) model to address these issues. We show that directly leveraging a 3D feature extractor combined with an effective feature-fusion scheme, and a simple cosine similarity for classification can yield better performance without the need of extra components for temporal modeling or complex distance functions. We introduce an innovative scheme to encode the textual semantics into the video representation which adaptively fuses features from text and video, and encourages the visual encoder to extract more semantically consistent features. In this scheme, SAFSAR achieves alignment and fusion in a compact way. Experiments on five challenging few-shot action recognition benchmarks under various settings demonstrate that the proposed SAFSAR model significantly improves the state-of-the-art performance.

Ambar Pal, Arnau Ramisa, Amit Kumar K C et al.

State-of-the-art object detectors are fast and accurate, but they require a large amount of well annotated training data to obtain good performance. However, obtaining a large amount of training annotations specific to a particular task, i.e., fine-grained annotations, is costly in practice. In contrast, obtaining common-sense relationships from text, e.g., "a table-lamp is a lamp that sits on top of a table", is much easier. Additionally, common-sense relationships like "on-top-of" are easy to annotate in a task-agnostic fashion. In this paper, we propose a probabilistic model that uses such relational knowledge to transform an off-the-shelf detector of coarse object categories (e.g., "table", "lamp") into a detector of fine-grained categories (e.g., "table-lamp"). We demonstrate that our method, RelDetect, achieves performance competitive to finetuning based state-of-the-art object detector baselines when an extremely low amount of fine-grained annotations is available ($0.2\%$ of entire dataset). We also demonstrate that RelDetect is able to utilize the inherent transferability of relationship information to obtain a better performance ($+5$ mAP points) than the above baselines on an unseen dataset (zero-shot transfer). In summary, we demonstrate the power of using relationships for object detection on datasets where fine-grained object categories can be linked to coarse-grained categories via suitable relationships.

Kwan Ho Ryan Chan, Aditya Chattopadhyay, Benjamin David Haeffele et al.

Variational Information Pursuit (V-IP) is a framework for making interpretable predictions by design by sequentially selecting a short chain of task-relevant, user-defined and interpretable queries about the data that are most informative for the task. While this allows for built-in interpretability in predictive models, applying V-IP to any task requires data samples with dense concept-labeling by domain experts, limiting the application of V-IP to small-scale tasks where manual data annotation is feasible. In this work, we extend the V-IP framework with Foundational Models (FMs) to address this limitation. More specifically, we use a two-step process, by first leveraging Large Language Models (LLMs) to generate a sufficiently large candidate set of task-relevant interpretable concepts, then using Large Multimodal Models to annotate each data sample by semantic similarity with each concept in the generated concept set. While other interpretable-by-design frameworks such as Concept Bottleneck Models (CBMs) require an additional step of removing repetitive and non-discriminative concepts to have good interpretability and test performance, we mathematically and empirically justify that, with a sufficiently informative and task-relevant query (concept) set, the proposed FM+V-IP method does not require any type of concept filtering. In addition, we show that FM+V-IP with LLM generated concepts can achieve better test performance than V-IP with human annotated concepts, demonstrating the effectiveness of LLMs at generating efficient query sets. Finally, when compared to other interpretable-by-design frameworks such as CBMs, FM+V-IP can achieve competitive test performance using fewer number of concepts/queries in both cases with filtered or unfiltered concept sets.

Darshan Thaker, Paris Giampouras, René Vidal

An important aspect of developing reliable deep learning systems is devising strategies that make these systems robust to adversarial attacks. There is a long line of work that focuses on developing defenses against these attacks, but recently, researchers have began to study ways to reverse engineer the attack process. This allows us to not only defend against several attack models, but also classify the threat model. However, there is still a lack of theoretical guarantees for the reverse engineering process. Current approaches that give any guarantees are based on the assumption that the data lies in a union of linear subspaces, which is not a valid assumption for more complex datasets. In this paper, we build on prior work and propose a novel framework for reverse engineering of deceptions which supposes that the clean data lies in the range of a GAN. To classify the signal and attack, we jointly solve a GAN inversion problem and a block-sparse recovery problem. For the first time in the literature, we provide deterministic linear convergence guarantees for this problem. We also empirically demonstrate the merits of the proposed approach on several nonlinear datasets as compared to state-of-the-art methods.

Buyun Liang, Kwan Ho Ryan Chan, Darshan Thaker et al.

Jailbreak attacks exploit specific prompts to bypass LLM safeguards, causing the LLM to generate harmful, inappropriate, and misaligned content. Current jailbreaking methods rely heavily on carefully designed system prompts and numerous queries to achieve a single successful attack, which is costly and impractical for large-scale red-teaming. To address this challenge, we propose to distill the knowledge of an ensemble of SOTA attackers into a single open-source model, called Knowledge-Distilled Attacker (KDA), which is finetuned to automatically generate coherent and diverse attack prompts without the need for meticulous system prompt engineering. Compared to existing attackers, KDA achieves higher attack success rates and greater cost-time efficiency when targeting multiple SOTA open-source and commercial black-box LLMs. Furthermore, we conducted a quantitative diversity analysis of prompts generated by baseline methods and KDA, identifying diverse and ensemble attacks as key factors behind KDA's effectiveness and efficiency.

Yashar Deldjoo, Zhankui He, Julian McAuley et al.

Traditional recommender systems (RS) typically use user-item rating histories as their main data source. However, deep generative models now have the capability to model and sample from complex data distributions, including user-item interactions, text, images, and videos, enabling novel recommendation tasks. This comprehensive, multidisciplinary survey connects key advancements in RS using Generative Models (Gen-RecSys), covering: interaction-driven generative models; the use of large language models (LLM) and textual data for natural language recommendation; and the integration of multimodal models for generating and processing images/videos in RS. Our work highlights necessary paradigms for evaluating the impact and harm of Gen-RecSys and identifies open challenges. This survey accompanies a tutorial presented at ACM KDD'24, with supporting materials provided at: https://encr.pw/vDhLq.

Buyun Liang, Liangzu Peng, Jinqi Luo et al.

Large Language Models (LLMs) are increasingly deployed in high-risk domains. However, state-of-the-art LLMs often produce hallucinations, raising serious concerns about their reliability. Prior work has explored adversarial attacks for hallucination elicitation in LLMs, but it often produces unrealistic prompts, either by inserting gibberish tokens or by altering the original meaning. As a result, these approaches offer limited insight into how hallucinations may occur in practice. While adversarial attacks in computer vision often involve realistic modifications to input images, the problem of finding realistic adversarial prompts for eliciting LLM hallucinations has remained largely underexplored. To address this gap, we propose Semantically Equivalent and Coherent Attacks (SECA) to elicit hallucinations via realistic modifications to the prompt that preserve its meaning while maintaining semantic coherence. Our contributions are threefold: (i) we formulate finding realistic attacks for hallucination elicitation as a constrained optimization problem over the input prompt space under semantic equivalence and coherence constraints; (ii) we introduce a constraint-preserving zeroth-order method to effectively search for adversarial yet feasible prompts; and (iii) we demonstrate through experiments on open-ended multiple-choice question answering tasks that SECA achieves higher attack success rates while incurring almost no constraint violations compared to existing methods. SECA highlights the sensitivity of both open-source and commercial gradient-inaccessible LLMs to realistic and plausible prompt variations. Code is available at https://github.com/Buyun-Liang/SECA.

Shangzhi Zhang, Chong You, René Vidal et al.

State-of-the-art subspace clustering methods are based on self-expressive model, which represents each data point as a linear combination of other data points. However, such methods are designed for a finite sample dataset and lack the ability to generalize to out-of-sample data. Moreover, since the number of self-expressive coefficients grows quadratically with the number of data points, their ability to handle large-scale datasets is often limited. In this paper, we propose a novel framework for subspace clustering, termed Self-Expressive Network (SENet), which employs a properly designed neural network to learn a self-expressive representation of the data. We show that our SENet can not only learn the self-expressive coefficients with desired properties on the training data, but also handle out-of-sample data. Besides, we show that SENet can also be leveraged to perform subspace clustering on large-scale datasets. Extensive experiments conducted on synthetic data and real world benchmark data validate the effectiveness of the proposed method. In particular, SENet yields highly competitive performance on MNIST, Fashion MNIST and Extended MNIST and state-of-the-art performance on CIFAR-10. The code is available at https://github.com/zhangsz1998/Self-Expressive-Network.

Tianyu Huang, Liangzu Peng, René Vidal et al.

Given an input set of $3$D point pairs, the goal of outlier-robust $3$D registration is to compute some rotation and translation that align as many point pairs as possible. This is an important problem in computer vision, for which many highly accurate approaches have been recently proposed. Despite their impressive performance, these approaches lack scalability, often overflowing the $16$GB of memory of a standard laptop to handle roughly $30,000$ point pairs. In this paper, we propose a $3$D registration approach that can process more than ten million ($10^7$) point pairs with over $99\%$ random outliers. Moreover, our method is efficient, entails low memory costs, and maintains high accuracy at the same time. We call our method TEAR, as it involves minimizing an outlier-robust loss that computes Truncated Entry-wise Absolute Residuals. To minimize this loss, we decompose the original $6$-dimensional problem into two subproblems of dimensions $3$ and $2$, respectively, solved in succession to global optimality via a customized branch-and-bound method. While branch-and-bound is often slow and unscalable, this does not apply to TEAR as we propose novel bounding functions that are tight and computationally efficient. Experiments on various datasets are conducted to validate the scalability and efficiency of our method.

Stefan Kolek, Aditya Chattopadhyay, Kwan Ho Ryan Chan et al.

Information Pursuit (IP) is an explainable prediction algorithm that greedily selects a sequence of interpretable queries about the data in order of information gain, updating its posterior at each step based on observed query-answer pairs. The standard paradigm uses hand-crafted dictionaries of potential data queries curated by a domain expert or a large language model after a human prompt. However, in practice, hand-crafted dictionaries are limited by the expertise of the curator and the heuristics of prompt engineering. This paper introduces a novel approach: learning a dictionary of interpretable queries directly from the dataset. Our query dictionary learning problem is formulated as an optimization problem by augmenting IP's variational formulation with learnable dictionary parameters. To formulate learnable and interpretable queries, we leverage the latent space of large vision and language models like CLIP. To solve the optimization problem, we propose a new query dictionary learning algorithm inspired by classical sparse dictionary learning. Our experiments demonstrate that learned dictionaries significantly outperform hand-crafted dictionaries generated with large language models.

Hancheng Min, René Vidal

The implicit bias of gradient-based training algorithms has been considered mostly beneficial as it leads to trained networks that often generalize well. However, Frei et al. (2023) show that such implicit bias can harm adversarial robustness. Specifically, they show that if the data consists of clusters with small inter-cluster correlation, a shallow (two-layer) ReLU network trained by gradient flow generalizes well, but it is not robust to adversarial attacks of small radius. Moreover, this phenomenon occurs despite the existence of a much more robust classifier that can be explicitly constructed from a shallow network. In this paper, we extend recent analyses of neuron alignment to show that a shallow network with a polynomial ReLU activation (pReLU) trained by gradient flow not only generalizes well but is also robust to adversarial attacks. Our results highlight the importance of the interplay between data structure and architecture design in the implicit bias and robustness of trained networks.

Uday Kiran Reddy Tadipatri, Benjamin D. Haeffele, Joshua Agterberg et al.

We propose a general framework for deriving generalization bounds for parallel positively homogeneous neural networks--a class of neural networks whose input-output map decomposes as the sum of positively homogeneous maps. Examples of such networks include matrix factorization and sensing, single-layer multi-head attention mechanisms, tensor factorization, deep linear and ReLU networks, and more. Our general framework is based on linking the non-convex empirical risk minimization (ERM) problem to a closely related convex optimization problem over prediction functions, which provides a global, achievable lower-bound to the ERM problem. We exploit this convex lower-bound to perform generalization analysis in the convex space while controlling the discrepancy between the convex model and its non-convex counterpart. We apply our general framework to a wide variety of models ranging from low-rank matrix sensing, to structured matrix sensing, two-layer linear networks, two-layer ReLU networks, and single-layer multi-head attention mechanisms, achieving generalization bounds with a sample complexity that scales almost linearly with the network width.

Jinqi Luo, Tianjiao Ding, Kwan Ho Ryan Chan et al.

Diffusion models are widely used for image editing tasks. Existing editing methods often design a representation manipulation procedure by curating an edit direction in the text embedding or score space. However, such a procedure faces a key challenge: overestimating the edit strength harms visual consistency while underestimating it fails the editing task. Notably, each source image may require a different editing strength, and it is costly to search for an appropriate strength via trial-and-error. To address this challenge, we propose Concept Lancet (CoLan), a zero-shot plug-and-play framework for principled representation manipulation in diffusion-based image editing. At inference time, we decompose the source input in the latent (text embedding or diffusion score) space as a sparse linear combination of the representations of the collected visual concepts. This allows us to accurately estimate the presence of concepts in each image, which informs the edit. Based on the editing task (replace/add/remove), we perform a customized concept transplant process to impose the corresponding editing direction. To sufficiently model the concept space, we curate a conceptual representation dataset, CoLan-150K, which contains diverse descriptions and scenarios of visual terms and phrases for the latent dictionary. Experiments on multiple diffusion-based image editing baselines show that methods equipped with CoLan achieve state-of-the-art performance in editing effectiveness and consistency preservation.

Kaleab A. Kinfu, Carolina Pacheco, Alice D. Sperry et al.

Motor imitation impairments are commonly reported in individuals with autism spectrum conditions (ASCs), suggesting that motor imitation could be used as a phenotype for addressing autism heterogeneity. Traditional methods for assessing motor imitation are subjective, labor-intensive, and require extensive human training. Modern Computerized Assessment of Motor Imitation (CAMI) methods, such as CAMI-3D for motion capture data and CAMI-2D for video data, are less subjective. However, they rely on labor-intensive data normalization and cleaning techniques, and human annotations for algorithm training. To address these challenges, we propose CAMI-2DNet, a scalable and interpretable deep learning-based approach to motor imitation assessment in video data, which eliminates the need for data normalization, cleaning and annotation. CAMI-2DNet uses an encoder-decoder architecture to map a video to a motion encoding that is disentangled from nuisance factors such as body shape and camera views. To learn a disentangled representation, we employ synthetic data generated by motion retargeting of virtual characters through the reshuffling of motion, body shape, and camera views, as well as real participant data. To automatically assess how well an individual imitates an actor, we compute a similarity score between their motion encodings, and use it to discriminate individuals with ASCs from neurotypical (NT) individuals. Our comparative analysis demonstrates that CAMI-2DNet has a strong correlation with human scores while outperforming CAMI-2D in discriminating ASC vs NT children. Moreover, CAMI-2DNet performs comparably to CAMI-3D while offering greater practicality by operating directly on video data and without the need for ad-hoc data normalization and human annotations.

Darshan Thaker, Abhishek Goyal, René Vidal

Image restoration aims to recover high-quality images from degraded observations. When the degradation process is known, the recovery problem can be formulated as an inverse problem, and in a Bayesian context, the goal is to sample a clean reconstruction given the degraded observation. Recently, modern pretrained diffusion models have been used for image restoration by modifying their sampling procedure to account for the degradation process. However, these methods often rely on certain approximations that can lead to significant errors and compromised sample quality. In this paper, we provide the first rigorous analysis of this approximation error for linear inverse problems under distributional assumptions on the space of natural images, demonstrating cases where previous works can fail dramatically. Motivated by our theoretical insights, we propose a simple modification to existing diffusion-based restoration methods. Our approach introduces a time-varying low-pass filter in the frequency domain of the measurements, progressively incorporating higher frequencies during the restoration process. We develop an adaptive curriculum for this frequency schedule based on the underlying data distribution. Our method significantly improves performance on challenging image restoration tasks including motion deblurring and image dehazing.

Hancheng Min, Zhihui Zhu, René Vidal

Among many mysteries behind the success of deep networks lies the exceptional discriminative power of their learned representations as manifested by the intriguing Neural Collapse (NC) phenomenon, where simple feature structures emerge at the last layer of a trained neural network. Prior works on the theoretical understandings of NC have focused on analyzing the optimization landscape of matrix-factorization-like problems by considering the last-layer features as unconstrained free optimization variables and showing that their global minima exhibit NC. In this paper, we show that gradient flow on a two-layer ReLU network for classifying orthogonally separable data provably exhibits NC, thereby advancing prior results in two ways: First, we relax the assumption of unconstrained features, showing the effect of data structure and nonlinear activations on NC characterizations. Second, we reveal the role of the implicit bias of the training dynamics in facilitating the emergence of NC.

Uday Kiran Reddy Tadipatri, Benjamin D. Haeffele, Joshua Agterberg et al.

Low-order linear System IDentification (SysID) addresses the challenge of estimating the parameters of a linear dynamical system from finite samples of observations and control inputs with minimal state representation. Traditional approaches often utilize Hankel-rank minimization, which relies on convex relaxations that can require numerous, costly singular value decompositions (SVDs) to optimize. In this work, we propose two nonconvex reformulations to tackle low-order SysID (i) Burer-Monterio (BM) factorization of the Hankel matrix for efficient nuclear norm minimization, and (ii) optimizing directly over system parameters for real, diagonalizable systems with an atomic norm style decomposition. These reformulations circumvent the need for repeated heavy SVD computations, significantly improving computational efficiency. Moreover, we prove that optimizing directly over the system parameters yields lower statistical error rates, and lower sample complexities that do not scale linearly with trajectory length like in Hankel-nuclear norm minimization. Additionally, while our proposed formulations are nonconvex, we provide theoretical guarantees of achieving global optimality in polynomial time. Finally, we demonstrate algorithms that solve these nonconvex programs and validate our theoretical claims on synthetic data.

Liangzu Peng, René Vidal

Continual learning is an emerging subject in machine learning that aims to solve multiple tasks presented sequentially to the learner without forgetting previously learned tasks. Recently, many deep learning based approaches have been proposed for continual learning, however the mathematical foundations behind existing continual learning methods remain underdeveloped. On the other hand, adaptive filtering is a classic subject in signal processing with a rich history of mathematically principled methods. However, its role in understanding the foundations of continual learning has been underappreciated. In this tutorial, we review the basic principles behind both continual learning and adaptive filtering, and present a comparative analysis that highlights multiple connections between them. These connections allow us to enhance the mathematical foundations of continual learning based on existing results for adaptive filtering, extend adaptive filtering insights using existing continual learning methods, and discuss a few research directions for continual learning suggested by the historical developments in adaptive filtering.

Kaleab A. Kinfu, René Vidal

Convolutional Neural Networks (CNNs) and Vision Transformers (ViTs) have led to significant progress in 2D body pose estimation. However, achieving a good balance between accuracy, efficiency, and robustness remains a challenge. For instance, CNNs are computationally efficient but struggle with long-range dependencies, while ViTs excel in capturing such dependencies but suffer from quadratic computational complexity. This paper proposes two ViT-based models for accurate, efficient, and robust 2D pose estimation. The first one, EViTPose, operates in a computationally efficient manner without sacrificing accuracy by utilizing learnable joint tokens to select and process a subset of the most important body patches, enabling us to control the trade-off between accuracy and efficiency by changing the number of patches to be processed. The second one, UniTransPose, while not allowing for the same level of direct control over the trade-off, efficiently handles multiple scales by combining (1) an efficient multi-scale transformer encoder that uses both local and global attention with (2) an efficient sub-pixel CNN decoder for better speed and accuracy. Moreover, by incorporating all joints from different benchmarks into a unified skeletal representation, we train robust methods that learn from multiple datasets simultaneously and perform well across a range of scenarios -- including pose variations, lighting conditions, and occlusions. Experiments on six benchmarks demonstrate that the proposed methods significantly outperform state-of-the-art methods while improving computational efficiency. EViTPose exhibits a significant decrease in computational complexity (30% to 44% less in GFLOPs) with a minimal drop of accuracy (0% to 3.5% less), and UniTransPose achieves accuracy improvements ranging from 0.9% to 43.8% across these benchmarks.

Ramchandran Muthukumar, Ambar Pal, Jeremias Sulam et al.

State-of-the-art machine learning systems are vulnerable to small perturbations to their input, where ``small'' is defined according to a threat model that assigns a positive threat to each perturbation. Most prior works define a task-agnostic, isotropic, and global threat, like the $\ell_p$ norm, where the magnitude of the perturbation fully determines the degree of the threat and neither the direction of the attack nor its position in space matter. However, common corruptions in computer vision, such as blur, compression, or occlusions, are not well captured by such threat models. This paper proposes a novel threat model called \texttt{Projected Displacement} (PD) to study robustness beyond existing isotropic and global threat models. The proposed threat model measures the threat of a perturbation via its alignment with \textit{unsafe directions}, defined as directions in the input space along which a perturbation of sufficient magnitude changes the ground truth class label. Unsafe directions are identified locally for each input based on observed training data. In this way, the PD threat model exhibits anisotropy and locality. Experiments on Imagenet-1k data indicate that, for any input, the set of perturbations with small PD threat includes \textit{safe} perturbations of large $\ell_p$ norm that preserve the true label, such as noise, blur and compression, while simultaneously excluding \textit{unsafe} perturbations that alter the true label. Unlike perceptual threat models based on embeddings of large-vision models, the PD threat model can be readily computed for arbitrary classification tasks without pre-training or finetuning. Further additional task annotation such as sensitivity to image regions or concept hierarchies can be easily integrated into the assessment of threat and thus the PD threat model presents practitioners with a flexible, task-driven threat specification.

Ambar Pal, René Vidal, Jeremias Sulam

Recent work in adversarial robustness suggests that natural data distributions are localized, i.e., they place high probability in small volume regions of the input space, and that this property can be utilized for designing classifiers with improved robustness guarantees for $\ell_2$-bounded perturbations. Yet, it is still unclear if this observation holds true for more general metrics. In this work, we extend this theory to $\ell_0$-bounded adversarial perturbations, where the attacker can modify a few pixels of the image but is unrestricted in the magnitude of perturbation, and we show necessary and sufficient conditions for the existence of $\ell_0$-robust classifiers. Theoretical certification approaches in this regime essentially employ voting over a large ensemble of classifiers. Such procedures are combinatorial and expensive or require complicated certification techniques. In contrast, a simple classifier emerges from our theory, dubbed Box-NN, which naturally incorporates the geometry of the problem and improves upon the current state-of-the-art in certified robustness against sparse attacks for the MNIST and Fashion-MNIST datasets.

Jinqi Luo, Tianjiao Ding, Kwan Ho Ryan Chan et al.

Large Language Models (LLMs) are being used for a wide variety of tasks. While they are capable of generating human-like responses, they can also produce undesirable output including potentially harmful information, racist or sexist language, and hallucinations. Alignment methods are designed to reduce such undesirable outputs via techniques such as fine-tuning, prompt engineering, and representation engineering. However, existing methods face several challenges: some require costly fine-tuning for every alignment task; some do not adequately remove undesirable concepts, failing alignment; some remove benign concepts, lowering the linguistic capabilities of LLMs. To address these issues, we propose Parsimonious Concept Engineering (PaCE), a novel activation engineering framework for alignment. First, to sufficiently model the concepts, we construct a large-scale concept dictionary in the activation space, in which each atom corresponds to a semantic concept. Given any alignment task, we instruct a concept partitioner to efficiently annotate the concepts as benign or undesirable. Then, at inference time, we decompose the LLM activations along the concept dictionary via sparse coding, to accurately represent the activations as linear combinations of benign and undesirable components. By removing the latter ones from the activations, we reorient the behavior of the LLM towards the alignment goal. We conduct experiments on tasks such as response detoxification, faithfulness enhancement, and sentiment revising, and show that PaCE achieves state-of-the-art alignment performance while maintaining linguistic capabilities.

Paris V. Giampouras, Benjamin D. Haeffele, René Vidal

Robust subspace recovery (RSR) is a fundamental problem in robust representation learning. Here we focus on a recently proposed RSR method termed Dual Principal Component Pursuit (DPCP) approach, which aims to recover a basis of the orthogonal complement of the subspace and is amenable to handling subspaces of high relative dimension. Prior work has shown that DPCP can provably recover the correct subspace in the presence of outliers, as long as the true dimension of the subspace is known. We show that DPCP can provably solve RSR problems in the {\it unknown} subspace dimension regime, as long as orthogonality constraints -- adopted in previous DPCP formulations -- are relaxed and random initialization is used instead of spectral one. Namely, we propose a very simple algorithm based on running multiple instances of a projected sub-gradient descent method (PSGM), with each problem instance seeking to find one vector in the null space of the subspace. We theoretically prove that under mild conditions this approach will succeed with high probability. In particular, we show that 1) all of the problem instances will converge to a vector in the nullspace of the subspace and 2) the ensemble of problem instance solutions will be sufficiently diverse to fully span the nullspace of the subspace thus also revealing its true unknown codimension. We provide empirical results that corroborate our theoretical results and showcase the remarkable implicit rank regularization behavior of PSGM algorithm that allows us to perform RSR without being aware of the subspace dimension.

Yunchen Yang, Xinyue Zhang, Tianjiao Ding et al.

In this paper, we revisit the problem of local optimization in RANSAC. Once a so-far-the-best model has been found, we refine it via Dual Principal Component Pursuit (DPCP), a robust subspace learning method with strong theoretical support and efficient algorithms. The proposed DPCP-RANSAC has far fewer parameters than existing methods and is scalable. Experiments on estimating two-view homographies, fundamental and essential matrices, and three-view homographic tensors using large-scale datasets show that our approach consistently has higher accuracy than state-of-the-art alternatives.

Danny Weyns, Thomas Bäck, Renè Vidal et al.

Computing systems form the backbone of many aspects of our life, hence they are becoming as vital as water, electricity, and road infrastructures for our society. Yet, engineering long running computing systems that achieve their goals in ever-changing environments pose significant challenges. Currently, we can build computing systems that adjust or learn over time to match changes that were anticipated. However, dealing with unanticipated changes, such as anomalies, novelties, new goals or constraints, requires system evolution, which remains in essence a human-driven activity. Given the growing complexity of computing systems and the vast amount of highly complex data to process, this approach will eventually become unmanageable. To break through the status quo, we put forward a new paradigm for the design and operation of computing systems that we coin "lifelong computing." The paradigm starts from computing-learning systems that integrate computing/service modules and learning modules. Computing warehouses offer such computing elements together with data sheets and usage guides. When detecting anomalies, novelties, new goals or constraints, a lifelong computing system activates an evolutionary self-learning engine that runs online experiments to determine how the computing-learning system needs to evolve to deal with the changes, thereby changing its architecture and integrating new computing elements from computing warehouses as needed. Depending on the domain at hand, some activities of lifelong computing systems can be supported by humans. We motivate the need for lifelong computing with a future fish farming scenario, outline a blueprint architecture for lifelong computing systems, and highlight key research challenges to realise the vision of lifelong computing.

Hancheng Min, Salma Tarmoun, Rene Vidal et al.

Neural networks trained via gradient descent with random initialization and without any regularization enjoy good generalization performance in practice despite being highly overparametrized. A promising direction to explain this phenomenon is to study how initialization and overparametrization affect convergence and implicit bias of training algorithms. In this paper, we present a novel analysis of single-hidden-layer linear networks trained under gradient flow, which connects initialization, optimization, and overparametrization. Firstly, we show that the squared loss converges exponentially to its optimum at a rate that depends on the level of imbalance and the margin of the initialization. Secondly, we show that proper initialization constrains the dynamics of the network parameters to lie within an invariant set. In turn, minimizing the loss over this set leads to the min-norm solution. Finally, we show that large hidden layer width, together with (properly scaled) random initialization, ensures proximity to such an invariant set during training, allowing us to derive a novel non-asymptotic upper-bound on the distance between the trained network and the min-norm solution.

Derek Lim, René Vidal, Benjamin D. Haeffele

Many state-of-the-art subspace clustering methods follow a two-step process by first constructing an affinity matrix between data points and then applying spectral clustering to this affinity. Most of the research into these methods focuses on the first step of generating the affinity, which often exploits the self-expressive property of linear subspaces, with little consideration typically given to the spectral clustering step that produces the final clustering. Moreover, existing methods often obtain the final affinity that is used in the spectral clustering step by applying ad-hoc or arbitrarily chosen postprocessing steps to the affinity generated by a self-expressive clustering formulation, which can have a significant impact on the overall clustering performance. In this work, we unify these two steps by learning both a self-expressive representation of the data and an affinity matrix that is well-normalized for spectral clustering. In our proposed models, we constrain the affinity matrix to be doubly stochastic, which results in a principled method for affinity matrix normalization while also exploiting known benefits of doubly stochastic normalization in spectral clustering. We develop a general framework and derive two models: one that jointly learns the self-expressive representation along with the doubly stochastic affinity, and one that sequentially solves for one then the other. Furthermore, we leverage sparsity in the problem to develop a fast active-set method for the sequential solver that enables efficient computation on large datasets. Experiments show that our method achieves state-of-the-art subspace clustering performance on many common datasets in computer vision.

Benjamin D. Haeffele, Chong You, René Vidal

Subspace clustering is an unsupervised clustering technique designed to cluster data that is supported on a union of linear subspaces, with each subspace defining a cluster with dimension lower than the ambient space. Many existing formulations for this problem are based on exploiting the self-expressive property of linear subspaces, where any point within a subspace can be represented as linear combination of other points within the subspace. To extend this approach to data supported on a union of non-linear manifolds, numerous studies have proposed learning an embedding of the original data using a neural network which is regularized by a self-expressive loss function on the data in the embedded space to encourage a union of linear subspaces prior on the data in the embedded space. Here we show that there are a number of potential flaws with this approach which have not been adequately addressed in prior work. In particular, we show the model formulation is often ill-posed in that it can lead to a degenerate embedding of the data, which need not correspond to a union of subspaces at all and is poorly suited for clustering. We validate our theoretical results experimentally and also repeat prior experiments reported in the literature, where we conclude that a significant portion of the previously claimed performance benefits can be attributed to an ad-hoc post processing step rather than the deep subspace clustering model.

Chong You, Chi Li, Daniel P. Robinson et al.

Finding a small set of representatives from an unlabeled dataset is a core problem in a broad range of applications such as dataset summarization and information extraction. Classical exemplar selection methods such as $k$-medoids work under the assumption that the data points are close to a few cluster centroids, and cannot handle the case where data lie close to a union of subspaces. This paper proposes a new exemplar selection model that searches for a subset that best reconstructs all data points as measured by the $\ell_1$ norm of the representation coefficients. Geometrically, this subset best covers all the data points as measured by the Minkowski functional of the subset. To solve our model efficiently, we introduce a farthest first search algorithm that iteratively selects the worst represented point as an exemplar. When the dataset is drawn from a union of independent subspaces, our method is able to select sufficiently many representatives from each subspace. We further develop an exemplar based subspace clustering method that is robust to imbalanced data and efficient for large scale data. Moreover, we show that a classifier trained on the selected exemplars (when they are labeled) can correctly classify the rest of the data points.

Chong You, Chun-Guang Li, Daniel P. Robinson et al.

Subspace clustering methods based on expressing each data point as a linear combination of other data points have achieved great success in computer vision applications such as motion segmentation, face and digit clustering. In face clustering, the subspaces are linear and subspace clustering methods can be applied directly. In motion segmentation, the subspaces are affine and an additional affine constraint on the coefficients is often enforced. However, since affine subspaces can always be embedded into linear subspaces of one extra dimension, it is unclear if the affine constraint is really necessary. This paper shows, both theoretically and empirically, that when the dimension of the ambient space is high relative to the sum of the dimensions of the affine subspaces, the affine constraint has a negligible effect on clustering performance. Specifically, our analysis provides conditions that guarantee the correctness of affine subspace clustering methods both with and without the affine constraint, and shows that these conditions are satisfied for high-dimensional data. Underlying our analysis is the notion of affinely independent subspaces, which not only provides geometrically interpretable correctness conditions, but also clarifies the relationships between existing results for affine subspace clustering.

Qing Qu, Zhihui Zhu, Xiao Li et al.

The problem of finding the sparsest vector (direction) in a low dimensional subspace can be considered as a homogeneous variant of the sparse recovery problem, which finds applications in robust subspace recovery, dictionary learning, sparse blind deconvolution, and many other problems in signal processing and machine learning. However, in contrast to the classical sparse recovery problem, the most natural formulation for finding the sparsest vector in a subspace is usually nonconvex. In this paper, we overview recent advances on global nonconvex optimization theory for solving this problem, ranging from geometric analysis of its optimization landscapes, to efficient optimization algorithms for solving the associated nonconvex optimization problem, to applications in machine intelligence, representation learning, and imaging sciences. Finally, we conclude this review by pointing out several interesting open problems for future research.

Daniel P. Robinson, Rene Vidal, Chong You

Given an overcomplete dictionary $A$ and a signal $b = Ac^*$ for some sparse vector $c^*$ whose nonzero entries correspond to linearly independent columns of $A$, classical sparse signal recovery theory considers the problem of whether $c^*$ can be recovered as the unique sparsest solution to $b = A c$. It is now well-understood that such recovery is possible by practical algorithms when the dictionary $A$ is incoherent or restricted isometric. In this paper, we consider the more general case where $b$ lies in a subspace $\mathcal{S}_0$ spanned by a subset of linearly dependent columns of $A$, and the remaining columns are outside of the subspace. In this case, the sparsest representation may not be unique, and the dictionary may not be incoherent or restricted isometric. The goal is to have the representation $c$ correctly identify the subspace, i.e. the nonzero entries of $c$ should correspond to columns of $A$ that are in the subspace $\mathcal{S}_0$. Such a representation $c$ is called subspace-preserving, a key concept that has found important applications for learning low-dimensional structures in high-dimensional data. We present various geometric conditions that guarantee subspace-preserving recovery. Among them, the major results are characterized by the covering radius and the angular distance, which capture the distribution of points in the subspace and the similarity between points in the subspace and points outside the subspace, respectively. Importantly, these conditions do not require the dictionary to be incoherent or restricted isometric. By establishing that the subspace-preserving recovery problem and the classical sparse signal recovery problem are equivalent under common assumptions on the latter, we show that several of our proposed conditions are generalizations of some well-known conditions in the sparse signal recovery literature.