Vikas Singh

Xingjian Zhen, Zihang Meng, Rudrasis Chakraborty et al.

Comparing the functional behavior of neural network models, whether it is a single network over time or two (or more networks) during or post-training, is an essential step in understanding what they are learning (and what they are not), and for identifying strategies for regularization or efficiency improvements. Despite recent progress, e.g., comparing vision transformers to CNNs, systematic comparison of function, especially across different networks, remains difficult and is often carried out layer by layer. Approaches such as canonical correlation analysis (CCA) are applicable in principle, but have been sparingly used so far. In this paper, we revisit a (less widely known) from statistics, called distance correlation (and its partial variant), designed to evaluate correlation between feature spaces of different dimensions. We describe the steps necessary to carry out its deployment for large scale models -- this opens the door to a surprising array of applications ranging from conditioning one deep model w.r.t. another, learning disentangled representations as well as optimizing diverse models that would directly be more robust to adversarial attacks. Our experiments suggest a versatile regularizer (or constraint) with many advantages, which avoids some of the common difficulties one faces in such analyses. Code is at https://github.com/zhenxingjian/Partial_Distance_Correlation.

Ronak Mehta, Sourav Pal, Vikas Singh et al.

Recent legislation has led to interest in machine unlearning, i.e., removing specific training samples from a predictive model as if they never existed in the training dataset. Unlearning may also be required due to corrupted/adversarial data or simply a user's updated privacy requirement. For models which require no training (k-NN), simply deleting the closest original sample can be effective. But this idea is inapplicable to models which learn richer representations. Recent ideas leveraging optimization-based updates scale poorly with the model dimension d, due to inverting the Hessian of the loss function. We use a variant of a new conditional independence coefficient, L-CODEC, to identify a subset of the model parameters with the most semantic overlap on an individual sample level. Our approach completely avoids the need to invert a (possibly) huge matrix. By utilizing a Markov blanket selection, we premise that L-CODEC is also suitable for deep unlearning, as well as other applications in vision. Compared to alternatives, L-CODEC makes approximate unlearning possible in settings that would otherwise be infeasible, including vision models used for face recognition, person re-identification and NLP models that may require unlearning samples identified for exclusion. Code can be found at https://github.com/vsingh-group/LCODEC-deep-unlearning/

Vishnu Suresh Lokhande, Rudrasis Chakraborty, Sathya N. Ravi et al.

Pooling multiple neuroimaging datasets across institutions often enables improvements in statistical power when evaluating associations (e.g., between risk factors and disease outcomes) that may otherwise be too weak to detect. When there is only a {\em single} source of variability (e.g., different scanners), domain adaptation and matching the distributions of representations may suffice in many scenarios. But in the presence of {\em more than one} nuisance variable which concurrently influence the measurements, pooling datasets poses unique challenges, e.g., variations in the data can come from both the acquisition method as well as the demographics of participants (gender, age). Invariant representation learning, by itself, is ill-suited to fully model the data generation process. In this paper, we show how bringing recent results on equivariant representation learning (for studying symmetries in neural networks) instantiated on structured spaces together with simple use of classical results on causal inference provides an effective practical solution. In particular, we demonstrate how our model allows dealing with more than one nuisance variable under some assumptions and can enable analysis of pooled scientific datasets in scenarios that would otherwise entail removing a large portion of the samples.

Zhanpeng Zeng, Michael Davies, Pranav Pulijala et al.

While GPU clusters are the de facto choice for training large deep neural network (DNN) models today, several reasons including ease of workflow, security and cost have led to efforts investigating whether CPUs may be viable for inference in routine use in many sectors of the industry. But the imbalance between the compute capabilities of GPUs and CPUs is huge. Motivated by these considerations, we study a module which is a workhorse within modern DNN architectures, GEMM based Feed Forward Networks (FFNs), and assess the extent to which it can be made compute- (or FLOP-) lite. Specifically, we propose an alternative formulation (we call it LookupFFN) to GEMM based FFNs inspired by the recent studies of using Locality Sensitive Hashing (LSH) to approximate FFNs. Our formulation recasts most essential operations as a memory look-up, leveraging the trade-off between the two resources on any platform: compute and memory (since CPUs offer it in abundance). For RoBERTa language model pretraining, our formulation achieves similar performance compared to GEMM based FFNs, while dramatically reducing the required FLOP. Our development is complemented with a detailed hardware profiling of strategies that will maximize efficiency -- not just on contemporary hardware but on products that will be offered in the near/medium term future. Code is avaiable at \url{https://github.com/mlpen/LookupFFN}.

Zhanpeng Zeng, Yunyang Xiong, Sathya N. Ravi et al.

Transformer-based models are widely used in natural language processing (NLP). Central to the transformer model is the self-attention mechanism, which captures the interactions of token pairs in the input sequences and depends quadratically on the sequence length. Training such models on longer sequences is expensive. In this paper, we show that a Bernoulli sampling attention mechanism based on Locality Sensitive Hashing (LSH), decreases the quadratic complexity of such models to linear. We bypass the quadratic cost by considering self-attention as a sum of individual tokens associated with Bernoulli random variables that can, in principle, be sampled at once by a single hash (although in practice, this number may be a small constant). This leads to an efficient sampling scheme to estimate self-attention which relies on specific modifications of LSH (to enable deployment on GPU architectures). We evaluate our algorithm on the GLUE benchmark with standard 512 sequence length where we see favorable performance relative to a standard pretrained Transformer. On the Long Range Arena (LRA) benchmark, for evaluating performance on long sequences, our method achieves results consistent with softmax self-attention but with sizable speed-ups and memory savings and often outperforms other efficient self-attention methods. Our code is available at https://github.com/mlpen/YOSO

Yunyang Xiong, Zhanpeng Zeng, Rudrasis Chakraborty et al.

Transformers have emerged as a powerful tool for a broad range of natural language processing tasks. A key component that drives the impressive performance of Transformers is the self-attention mechanism that encodes the influence or dependence of other tokens on each specific token. While beneficial, the quadratic complexity of self-attention on the input sequence length has limited its application to longer sequences -- a topic being actively studied in the community. To address this limitation, we propose Nyströmformer -- a model that exhibits favorable scalability as a function of sequence length. Our idea is based on adapting the Nyström method to approximate standard self-attention with $O(n)$ complexity. The scalability of Nyströmformer enables application to longer sequences with thousands of tokens. We perform evaluations on multiple downstream tasks on the GLUE benchmark and IMDB reviews with standard sequence length, and find that our Nyströmformer performs comparably, or in a few cases, even slightly better, than standard self-attention. On longer sequence tasks in the Long Range Arena (LRA) benchmark, Nyströmformer performs favorably relative to other efficient self-attention methods. Our code is available at https://github.com/mlpen/Nystromformer.

Sathya N. Ravi, Maxwell D. Collins, Vikas Singh

We present a new Frank-Wolfe (FW) type algorithm that is applicable to minimization problems with a nonsmooth convex objective. We provide convergence bounds and show that the scheme yields so-called coreset results for various Machine Learning problems including 1-median, Balanced Development, Sparse PCA, Graph Cuts, and the $\ell_1$-norm-regularized Support Vector Machine (SVM) among others. This means that the algorithm provides approximate solutions to these problems in time complexity bounds that are not dependent on the size of the input problem. Our framework, motivated by a growing body of work on sublinear algorithms for various data analysis problems, is entirely deterministic and makes no use of smoothing or proximal operators. Apart from these theoretical results, we show experimentally that the algorithm is very practical and in some cases also offers significant computational advantages on large problem instances. We provide an open source implementation that can be adapted for other problems that fit the overall structure.

Md Asifuzzaman Jishan, Vikas Singh, Ayan Kumar Ghosh et al.

This study applies Bayesian models to predict hotel booking cancellations, a key challenge affecting resource allocation, revenue, and customer satisfaction in the hospitality industry. Using a Kaggle dataset with 36,285 observations and 17 features, Bayesian Logistic Regression and Beta-Binomial models were implemented. The logistic model, applied to 12 features and 5,000 randomly selected observations, outperformed the Beta-Binomial model in predictive accuracy. Key predictors included the number of adults, children, stay duration, lead time, car parking space, room type, and special requests. Model evaluation using Leave-One-Out Cross-Validation (LOO-CV) confirmed strong alignment between observed and predicted outcomes, demonstrating the model's robustness. Special requests and parking availability were found to be the strongest predictors of cancellation. This Bayesian approach provides a valuable tool for improving booking management and operational efficiency in the hotel industry.

Sotirios Panagiotis Chytas, Vishnu Suresh Lokhande, Peiran Li et al.

Small sample sizes are common in many disciplines, which necessitates pooling roughly similar datasets across multiple institutions to study weak but relevant associations between images and disease outcomes. Such data often manifest shift/imbalance in covariates (i.e., secondary non-imaging data). Controlling for such nuisance variables is common within standard statistical analysis, but the ideas do not directly apply to overparameterized models. Consequently, recent work has shown how strategies from invariant representation learning provides a meaningful starting point, but the current repertoire of methods is limited to accounting for shifts/imbalances in just a couple of covariates at a time. In this paper, we show how viewing this problem from the perspective of Category theory provides a simple and effective solution that completely avoids elaborate multi-stage training pipelines that would otherwise be needed. We show the effectiveness of this approach via extensive experiments on real datasets. Further, we discuss how this style of formulation offers a unified perspective on at least 5+ distinct problem settings, from self-supervised learning to matching problems in 3D reconstruction.

Ziteng Sun, Adrian Benton, Samuel Kushnir et al.

Post-training quantization is an effective method for reducing the serving cost of large language models, where the standard approach is to use a round-to-nearest quantization level scheme. However, this often introduces large errors due to outliers in the weights. Proposed mitigation mechanisms include applying adaptive rounding, random rotation transformations or committing to a post-training target using calibration data. Unfortunately, this reliance on calibration data can be severely limiting in some real-world scenarios as such data may be unavailable or subject to privacy regulations. In this paper, we propose algorithms to optimize transformations and adaptive rounding without access to any calibration data. The optimization is achieved by designing a suitable proxy function for the quantization loss without calibration data. To maintain inference efficiency, we perform structured matrix transformations for single matrices. For paired weights that interact directly in the computation graph, we use dual matrix transformations and adaptive rounding methods. We conduct experiments on Gemma 2 models, and observe consistent improvement over the baselines. For Gemma 2 9B quantization, our method improves the average benchmark score from 61.9 to 62.4 for 4-bit quantization and from 52.0 to 60.6 for 3-bit quantization, while adding less than 3% of computation overhead. Furthermore, our method achieves performance comparable to the commonly used GPTQ method, which requires calibration data.

Jurijs Nazarovs, Zhichun Huang, Xingjian Zhen et al.

A deterministic temporal process can be determined by its trajectory, an element in the product space of (a) initial condition $z_0 \in \mathcal{Z}$ and (b) transition function $f: (\mathcal{Z}, \mathcal{T}) \to \mathcal{Z}$ often influenced by the control of the underlying dynamical system. Existing methods often model the transition function as a differential equation or as a recurrent neural network. Despite their effectiveness in predicting future measurements, few results have successfully established a method for sampling and statistical inference of trajectories using neural networks, partially due to constraints in the parameterization. In this work, we introduce a mechanism to learn the distribution of trajectories by parameterizing the transition function $f$ explicitly as an element in a function space. Our framework allows efficient synthesis of novel trajectories, while also directly providing a convenient tool for inference, i.e., uncertainty estimation, likelihood evaluations and out of distribution detection for abnormal trajectories. These capabilities can have implications for various downstream tasks, e.g., simulation and evaluation for reinforcement learning.

Zhanpeng Zeng, Cole Hawkins, Mingyi Hong et al.

Transformers are central in modern natural language processing and computer vision applications. Despite recent works devoted to reducing the quadratic cost of such models (as a function of the sequence length), dealing with ultra long sequences (e.g., with more than 16K tokens) remains challenging. Applications such as answering questions based on a book or summarizing a scientific article are inefficient or infeasible. Here, we propose to significantly improve the efficiency of Transformers for ultra long sequences, by compressing the sequence into a much smaller representation at each layer. Specifically, by exploiting the fact that in many tasks, only a small subset of special tokens (we call VIP-tokens) are most relevant to the final prediction, we propose a VIP-token centric compression (VCC) scheme which selectively compresses the sequence based on their impact on approximating the representation of the VIP-tokens. Compared with competitive baselines, our algorithm is not only efficient (achieving more than $3\times$ efficiency gain compared to baselines on 4K and 16K lengths), but also offers competitive/better performance on a large number of tasks. Further, we show that our algorithm scales to 128K tokens (or more) while consistently offering accuracy improvement.

Jurijs Nazarovs, Ronak R. Mehta, Vishnu Suresh Lokhande et al.

Uncertainty estimation in deep models is essential in many real-world applications and has benefited from developments over the last several years. Recent evidence suggests that existing solutions dependent on simple Gaussian formulations may not be sufficient. However, moving to other distributions necessitates Monte Carlo (MC) sampling to estimate quantities such as the KL divergence: it could be expensive and scales poorly as the dimensions of both the input data and the model grow. This is directly related to the structure of the computation graph, which can grow linearly as a function of the number of MC samples needed. Here, we construct a framework to describe these computation graphs, and identify probability families where the graph size can be independent or only weakly dependent on the number of MC samples. These families correspond directly to large classes of distributions. Empirically, we can run a much larger number of iterations for MC approximations for larger architectures used in computer vision with gains in performance measured in confident accuracy, stability of training, memory and training time.

Jurijs Nazarovs, Rudrasis Chakraborty, Songwong Tasneeyapant et al.

Panel data involving longitudinal measurements of the same set of participants taken over multiple time points is common in studies to understand childhood development and disease modeling. Deep hybrid models that marry the predictive power of neural networks with physical simulators such as differential equations, are starting to drive advances in such applications. The task of modeling not just the observations but the hidden dynamics that are captured by the measurements poses interesting statistical/computational questions. We propose a probabilistic model called ME-NODE to incorporate (fixed + random) mixed effects for analyzing such panel data. We show that our model can be derived using smooth approximations of SDEs provided by the Wong-Zakai theorem. We then derive Evidence Based Lower Bounds for ME-NODE, and develop (efficient) training algorithms using MC based sampling methods and numerical ODE solvers. We demonstrate ME-NODE's utility on tasks spanning the spectrum from simulations and toy data to real longitudinal 3D imaging data from an Alzheimer's disease (AD) study, and study its performance in terms of accuracy of reconstruction for interpolation, uncertainty estimates and personalized prediction.

Zhichun Huang, Rudrasis Chakraborty, Vikas Singh

Generative models which use explicit density modeling (e.g., variational autoencoders, flow-based generative models) involve finding a mapping from a known distribution, e.g. Gaussian, to the unknown input distribution. This often requires searching over a class of non-linear functions (e.g., representable by a deep neural network). While effective in practice, the associated runtime/memory costs can increase rapidly, usually as a function of the performance desired in an application. We propose a much cheaper (and simpler) strategy to estimate this mapping based on adapting known results in kernel transfer operators. We show that our formulation enables highly efficient distribution approximation and sampling, and offers surprisingly good empirical performance that compares favorably with powerful baselines, but with significant runtime savings. We show that the algorithm also performs well in small sample size settings (in brain imaging).

Xingjian Zhen, Rudrasis Chakraborty, Vikas Singh

One strategy for adversarially training a robust model is to maximize its certified radius -- the neighborhood around a given training sample for which the model's prediction remains unchanged. The scheme typically involves analyzing a "smoothed" classifier where one estimates the prediction corresponding to Gaussian samples in the neighborhood of each sample in the mini-batch, accomplished in practice by Monte Carlo sampling. In this paper, we investigate the hypothesis that this sampling bottleneck can potentially be mitigated by identifying ways to directly propagate the covariance matrix of the smoothed distribution through the network. To this end, we find that other than certain adjustments to the network, propagating the covariances must also be accompanied by additional accounting that keeps track of how the distributional moments transform and interact at each stage in the network. We show how satisfying these criteria yields an algorithm for maximizing the certified radius on datasets including Cifar-10, ImageNet, and Places365 while offering runtime savings on networks with moderate depth, with a small compromise in overall accuracy. We describe the details of the key modifications that enable practical use. Via various experiments, we evaluate when our simplifications are sensible, and what the key benefits and limitations are.

Aditya Kumar Akash, Vishnu Suresh Lokhande, Sathya N. Ravi et al.

Learning invariant representations is a critical first step in a number of machine learning tasks. A common approach corresponds to the so-called information bottleneck principle in which an application dependent function of mutual information is carefully chosen and optimized. Unfortunately, in practice, these functions are not suitable for optimization purposes since these losses are agnostic of the metric structure of the parameters of the model. We introduce a class of losses for learning representations that are invariant to some extraneous variable of interest by inverting the class of contrastive losses, i.e., inverse contrastive loss (ICL). We show that if the extraneous variable is binary, then optimizing ICL is equivalent to optimizing a regularized MMD divergence. More generally, we also show that if we are provided a metric on the sample space, our formulation of ICL can be decomposed into a sum of convex functions of the given distance metric. Our experimental results indicate that models obtained by optimizing ICL achieve significantly better invariance to the extraneous variable for a fixed desired level of accuracy. In a variety of experimental settings, we show applicability of ICL for learning invariant representations for both continuous and discrete extraneous variables.

Xingjian Zhen, Rudrasis Chakraborty, Liu Yang et al.

Many measurements or observations in computer vision and machine learning manifest as non-Euclidean data. While recent proposals (like spherical CNN) have extended a number of deep neural network architectures to manifold-valued data, and this has often provided strong improvements in performance, the literature on generative models for manifold data is quite sparse. Partly due to this gap, there are also no modality transfer/translation models for manifold-valued data whereas numerous such methods based on generative models are available for natural images. This paper addresses this gap, motivated by a need in brain imaging -- in doing so, we expand the operating range of certain generative models (as well as generative models for modality transfer) from natural images to images with manifold-valued measurements. Our main result is the design of a two-stream version of GLOW (flow-based invertible generative models) that can synthesize information of a field of one type of manifold-valued measurements given another. On the theoretical side, we introduce three kinds of invertible layers for manifold-valued data, which are not only analogous to their functionality in flow-based generative models (e.g., GLOW) but also preserve the key benefits (determinants of the Jacobian are easy to calculate). For experiments, on a large dataset from the Human Connectome Project (HCP), we show promising results where we can reliably and accurately reconstruct brain images of a field of orientation distribution functions (ODF) from diffusion tensor images (DTI), where the latter has a $5\times$ faster acquisition time but at the expense of worse angular resolution.

Zihang Meng, Sathya N. Ravi, Vikas Singh

Consider a learning algorithm, which involves an internal call to an optimization routine such as a generalized eigenvalue problem, a cone programming problem or even sorting. Integrating such a method as a layer(s) within a trainable deep neural network (DNN) in an efficient and numerically stable way is not straightforward -- for instance, only recently, strategies have emerged for eigendecomposition and differentiable sorting. We propose an efficient and differentiable solver for general linear programming problems which can be used in a plug and play manner within DNNs as a layer. Our development is inspired by a fascinating but not widely used link between dynamics of slime mold (physarum) and optimization schemes such as steepest descent. We describe our development and show the use of our solver in a video segmentation task and meta-learning for few-shot learning. We review the existing results and provide a technical analysis describing its applicability for our use cases. Our solver performs comparably with a customized projected gradient descent method on the first task and outperforms the differentiable CVXPY-SCS solver on the second task. Experiments show that our solver converges quickly without the need for a feasible initial point. Our proposal is easy to implement and can easily serve as layers whenever a learning procedure needs a fast approximate solution to a LP, within a larger network.

Vishnu Suresh Lokhande, Aditya Kumar Akash, Sathya N. Ravi et al.

Algorithmic decision making based on computer vision and machine learning technologies continue to permeate our lives. But issues related to biases of these models and the extent to which they treat certain segments of the population unfairly, have led to concern in the general public. It is now accepted that because of biases in the datasets we present to the models, a fairness-oblivious training will lead to unfair models. An interesting topic is the study of mechanisms via which the de novo design or training of the model can be informed by fairness measures. Here, we study mechanisms that impose fairness concurrently while training the model. While existing fairness based approaches in vision have largely relied on training adversarial modules together with the primary classification/regression task, in an effort to remove the influence of the protected attribute or variable, we show how ideas based on well-known optimization concepts can provide a simpler alternative. In our proposed scheme, imposing fairness just requires specifying the protected attribute and utilizing our optimization routine. We provide a detailed technical analysis and present experiments demonstrating that various fairness measures from the literature can be reliably imposed on a number of training tasks in vision in a manner that is interpretable.

Xingjian Zhen, Rudrasis Chakraborty, Nicholas Vogt et al.

Efforts are underway to study ways via which the power of deep neural networks can be extended to non-standard data types such as structured data (e.g., graphs) or manifold-valued data (e.g., unit vectors or special matrices). Often, sizable empirical improvements are possible when the geometry of such data spaces are incorporated into the design of the model, architecture, and the algorithms. Motivated by neuroimaging applications, we study formulations where the data are {\em sequential manifold-valued measurements}. This case is common in brain imaging, where the samples correspond to symmetric positive definite matrices or orientation distribution functions. Instead of a recurrent model which poses computational/technical issues, and inspired by recent results showing the viability of dilated convolutional models for sequence prediction, we develop a dilated convolutional neural network architecture for this task. On the technical side, we show how the modules needed in our network can be derived while explicitly taking the Riemannian manifold structure into account. We show how the operations needed can leverage known results for calculating the weighted Fréchet Mean (wFM). Finally, we present scientific results for group difference analysis in Alzheimer's disease (AD) where the groups are derived using AD pathology load: here the model finds several brain fiber bundles that are related to AD even when the subjects are all still cognitively healthy.

Sathya N. Ravi, Abhay Venkatesh, Glenn Moo Fung et al.

Data dependent regularization is known to benefit a wide variety of problems in machine learning. Often, these regularizers cannot be easily decomposed into a sum over a finite number of terms, e.g., a sum over individual example-wise terms. The $F_β$ measure, Area under the ROC curve (AUCROC) and Precision at a fixed recall (P@R) are some prominent examples that are used in many applications. We find that for most medium to large sized datasets, scalability issues severely limit our ability in leveraging the benefits of such regularizers. Importantly, the key technical impediment despite some recent progress is that, such objectives remain difficult to optimize via backpropapagation procedures. While an efficient general-purpose strategy for this problem still remains elusive, in this paper, we show that for many data-dependent nondecomposable regularizers that are relevant in applications, sizable gains in efficiency are possible with minimal code-level changes; in other words, no specialized tools or numerical schemes are needed. Our procedure involves a reparameterization followed by a partial dualization -- this leads to a formulation that has provably cheap projection operators. We present a detailed analysis of runtime and convergence properties of our algorithm. On the experimental side, we show that a direct use of our scheme significantly improves the state of the art IOU measures reported for MSCOCO Stuff segmentation dataset.

Vishnu Suresh Lokhande, Songwong Tasneeyapant, Abhay Venkatesh et al.

Rectified Linear Units (ReLUs) are among the most widely used activation function in a broad variety of tasks in vision. Recent theoretical results suggest that despite their excellent practical performance, in various cases, a substitution with basis expansions (e.g., polynomials) can yield significant benefits from both the optimization and generalization perspective. Unfortunately, the existing results remain limited to networks with a couple of layers, and the practical viability of these results is not yet known. Motivated by some of these results, we explore the use of Hermite polynomial expansions as a substitute for ReLUs in deep networks. While our experiments with supervised learning do not provide a clear verdict, we find that this strategy offers considerable benefits in semi-supervised learning (SSL) / transductive learning settings. We carefully develop this idea and show how the use of Hermite polynomials based activations can yield improvements in pseudo-label accuracies and sizable financial savings (due to concurrent runtime benefits). Further, we show via theoretical analysis, that the networks (with Hermite activations) offer robustness to noise and other attractive mathematical properties.

Rudrasis Chakraborty, Chun-Hao Yang, Xingjian Zhen et al.

In a number of disciplines, the data (e.g., graphs, manifolds) to be analyzed are non-Euclidean in nature. Geometric deep learning corresponds to techniques that generalize deep neural network models to such non-Euclidean spaces. Several recent papers have shown how convolutional neural networks (CNNs) can be extended to learn with graph-based data. In this work, we study the setting where the data (or measurements) are ordered, longitudinal or temporal in nature and live on a Riemannian manifold -- this setting is common in a variety of problems in statistical machine learning, vision and medical imaging. We show how recurrent statistical recurrent network models can be defined in such spaces. We give an efficient algorithm and conduct a rigorous analysis of its statistical properties. We perform extensive numerical experiments demonstrating competitive performance with state of the art methods but with significantly less number of parameters. We also show applications to a statistical analysis task in brain imaging, a regime where deep neural network models have only been utilized in limited ways.

Sathya N. Ravi, Tuan Dinh, Vishnu Lokhande et al.

A number of results have recently demonstrated the benefits of incorporating various constraints when training deep architectures in vision and machine learning. The advantages range from guarantees for statistical generalization to better accuracy to compression. But support for general constraints within widely used libraries remains scarce and their broader deployment within many applications that can benefit from them remains under-explored. Part of the reason is that Stochastic gradient descent (SGD), the workhorse for training deep neural networks, does not natively deal with constraints with global scope very well. In this paper, we revisit a classical first order scheme from numerical optimization, Conditional Gradients (CG), that has, thus far had limited applicability in training deep models. We show via rigorous analysis how various constraints can be naturally handled by modifications of this algorithm. We provide convergence guarantees and show a suite of immediate benefits that are possible -- from training ResNets with fewer layers but better accuracy simply by substituting in our version of CG to faster training of GANs with 50% fewer epochs in image inpainting applications to provably better generalization guarantees using efficiently implementable forms of recently proposed regularizers.

Ronak Mehta, Hyunwoo J. Kim, Shulei Wang et al.

Recent results in coupled or temporal graphical models offer schemes for estimating the relationship structure between features when the data come from related (but distinct) longitudinal sources. A novel application of these ideas is for analyzing group-level differences, i.e., in identifying if trends of estimated objects (e.g., covariance or precision matrices) are different across disparate conditions (e.g., gender or disease). Often, poor effect sizes make detecting the differential signal over the full set of features difficult: for example, dependencies between only a subset of features may manifest differently across groups. In this work, we first give a parametric model for estimating trends in the space of SPD matrices as a function of one or more covariates. We then generalize scan statistics to graph structures, to search over distinct subsets of features (graph partitions) whose temporal dependency structure may show statistically significant group-wise differences. We theoretically analyze the Family Wise Error Rate (FWER) and bounds on Type 1 and Type 2 error. On a cohort of individuals with risk factors for Alzheimer's disease (but otherwise cognitively healthy), we find scientifically interesting group differences where the default analysis, i.e., models estimated on the full graph, do not survive reasonable significance thresholds.

Hao Henry Zhou, Yilin Zhang, Vamsi K. Ithapu et al.

Many studies in biomedical and health sciences involve small sample sizes due to logistic or financial constraints. Often, identifying weak (but scientifically interesting) associations between a set of predictors and a response necessitates pooling datasets from multiple diverse labs or groups. While there is a rich literature in statistical machine learning to address distributional shifts and inference in multi-site datasets, it is less clear ${\it when}$ such pooling is guaranteed to help (and when it does not) -- independent of the inference algorithms we use. In this paper, we present a hypothesis test to answer this question, both for classical and high dimensional linear regression. We precisely identify regimes where pooling datasets across multiple sites is sensible, and how such policy decisions can be made via simple checks executable on each site before any data transfer ever happens. With a focus on Alzheimer's disease studies, we present empirical results showing that in regimes suggested by our analysis, pooling a local dataset with data from an international study improves power.

Vamsi K. Ithapu, Risi Kondor, Sterling C. Johnson et al.

Multiresolution analysis and matrix factorization are foundational tools in computer vision. In this work, we study the interface between these two distinct topics and obtain techniques to uncover hierarchical block structure in symmetric matrices -- an important aspect in the success of many vision problems. Our new algorithm, the incremental multiresolution matrix factorization, uncovers such structure one feature at a time, and hence scales well to large matrices. We describe how this multiscale analysis goes much farther than what a direct global factorization of the data can identify. We evaluate the efficacy of the resulting factorizations for relative leveraging within regression tasks using medical imaging data. We also use the factorization on representations learned by popular deep networks, providing evidence of their ability to infer semantic relationships even when they are not explicitly trained to do so. We show that this algorithm can be used as an exploratory tool to improve the network architecture, and within numerous other settings in vision.

Felipe Gutierrez-Barragan, Vamsi K. Ithapu, Chris Hinrichs et al.

Permutation testing is a non-parametric method for obtaining the max null distribution used to compute corrected $p$-values that provide strong control of false positives. In neuroimaging, however, the computational burden of running such an algorithm can be significant. We find that by viewing the permutation testing procedure as the construction of a very large permutation testing matrix, $T$, one can exploit structural properties derived from the data and the test statistics to reduce the runtime under certain conditions. In particular, we see that $T$ is low-rank plus a low-variance residual. This makes $T$ a good candidate for low-rank matrix completion, where only a very small number of entries of $T$ ($\sim0.35\%$ of all entries in our experiments) have to be computed to obtain a good estimate. Based on this observation, we present RapidPT, an algorithm that efficiently recovers the max null distribution commonly obtained through regular permutation testing in voxel-wise analysis. We present an extensive validation on a synthetic dataset and four varying sized datasets against two baselines: Statistical NonParametric Mapping (SnPM13) and a standard permutation testing implementation (referred as NaivePT). We find that RapidPT achieves its best runtime performance on medium sized datasets ($50 \leq n \leq 200$), with speedups of 1.5x - 38x (vs. SnPM13) and 20x-1000x (vs. NaivePT). For larger datasets ($n \geq 200$) RapidPT outperforms NaivePT (6x - 200x) on all datasets, and provides large speedups over SnPM13 when more than 10000 permutations (2x - 15x) are needed. The implementation is a standalone toolbox and also integrated within SnPM13, able to leverage multi-core architectures when available.

Vamsi K Ithapu, Sathya N Ravi, Vikas Singh

We study mechanisms to characterize how the asymptotic convergence of backpropagation in deep architectures, in general, is related to the network structure, and how it may be influenced by other design choices including activation type, denoising and dropout rate. We seek to analyze whether network architecture and input data statistics may guide the choices of learning parameters and vice versa. Given the broad applicability of deep architectures, this issue is interesting both from theoretical and a practical standpoint. Using properties of general nonconvex objectives (with first-order information), we first build the association between structural, distributional and learnability aspects of the network vis-à-vis their interaction with parameter convergence rates. We identify a nice relationship between feature denoising and dropout, and construct families of networks that achieve the same level of convergence. We then derive a workflow that provides systematic guidance regarding the choice of network sizes and learning parameters often mediated4 by input statistics. Our technical results are corroborated by an extensive set of evaluations, presented in this paper as well as independent empirical observations reported by other groups. We also perform experiments showing the practical implications of our framework for choosing the best fully-connected design for a given problem.

Vamsi K. Ithapu, Sathya Ravi, Vikas Singh

Unsupervised pretraining and dropout have been well studied, especially with respect to regularization and output consistency. However, our understanding about the explicit convergence rates of the parameter estimates, and their dependence on the learning (like denoising and dropout rate) and structural (like depth and layer lengths) aspects of the network is less mature. An interesting question in this context is to ask if the network structure could "guide" the choices of such learning parameters. In this work, we explore these gaps between network structure, the learning mechanisms and their interaction with parameter convergence rates. We present a way to address these issues based on the backpropagation convergence rates for general nonconvex objectives using first-order information. We then incorporate two learning mechanisms into this general framework -- denoising autoencoder and dropout, and subsequently derive the convergence rates of deep networks. Building upon these bounds, we provide insights into the choices of learning parameters and network sizes that achieve certain levels of convergence accuracy. The results derived here support existing empirical observations, and we also conduct a set of experiments to evaluate them.

Chris Hinrichs, Vamsi K Ithapu, Qinyuan Sun et al.

Multiple hypothesis testing is a significant problem in nearly all neuroimaging studies. In order to correct for this phenomena, we require a reliable estimate of the Family-Wise Error Rate (FWER). The well known Bonferroni correction method, while simple to implement, is quite conservative, and can substantially under-power a study because it ignores dependencies between test statistics. Permutation testing, on the other hand, is an exact, non-parametric method of estimating the FWER for a given $α$-threshold, but for acceptably low thresholds the computational burden can be prohibitive. In this paper, we show that permutation testing in fact amounts to populating the columns of a very large matrix ${\bf P}$. By analyzing the spectrum of this matrix, under certain conditions, we see that ${\bf P}$ has a low-rank plus a low-variance residual decomposition which makes it suitable for highly sub--sampled --- on the order of $0.5\%$ --- matrix completion methods. Based on this observation, we propose a novel permutation testing methodology which offers a large speedup, without sacrificing the fidelity of the estimated FWER. Our evaluations on four different neuroimaging datasets show that a computational speedup factor of roughly $50\times$ can be achieved while recovering the FWER distribution up to very high accuracy. Further, we show that the estimated $α$-threshold is also recovered faithfully, and is stable.

Deepti Pachauri, Maxwell Collins, Vikas SIngh et al.

Matching one set of objects to another is a ubiquitous task in machine learning and computer vision that often reduces to some form of the quadratic assignment problem (QAP). The QAP is known to be notoriously hard, both in theory and in practice. Here, we investigate if this difficulty can be mitigated when some additional piece of information is available: (a) that all QAP instances of interest come from the same application, and (b) the correct solution for a set of such QAP instances is given. We propose a new approach to accelerate the solution of QAPs based on learning parameters for a modified objective function from prior QAP instances. A key feature of our approach is that it takes advantage of the algebraic structure of permutations, in conjunction with special methods for optimizing functions over the symmetric group Sn in Fourier space. Experiments show that in practical domains the new method can outperform existing approaches.