Prateek Jain

Gagan Jain, Nidhi Hegde, Aditya Kusupati et al. · uw

The visual medium (images and videos) naturally contains a large amount of information redundancy, thereby providing a great opportunity for leveraging efficiency in processing. While Vision Transformer (ViT) based models scale effectively to large data regimes, they fail to capitalize on this inherent redundancy, leading to higher computational costs. Mixture of Experts (MoE) networks demonstrate scalability while maintaining same inference-time costs, but they come with a larger parameter footprint. We present Mixture of Nested Experts (MoNE), which utilizes a nested structure for experts, wherein individual experts fall on an increasing compute-accuracy curve. Given a compute budget, MoNE learns to dynamically choose tokens in a priority order, and thus redundant tokens are processed through cheaper nested experts. Using this framework, we achieve equivalent performance as the baseline models, while reducing inference time compute by over two-fold. We validate our approach on standard image and video datasets - ImageNet-21K, Kinetics400, and Something-Something-v2. We further highlight MoNE$'$s adaptability by showcasing its ability to maintain strong performance across different inference-time compute budgets on videos, using only a single trained model.

Abhishek Sharma, Arpit Jain, Shubhangi Sharma et al.

ADHD is a prevalent disorder among the younger population. Standard evaluation techniques currently use evaluation forms, interviews with the patient, and more. However, its symptoms are similar to those of many other disorders like depression, conduct disorder, and oppositional defiant disorder, and these current diagnosis techniques are not very effective. Thus, a sophisticated computing model holds the potential to provide a promising diagnosis solution to this problem. This work attempts to explore methods to diagnose ADHD using combinations of multiple established machine learning techniques like neural networks and SVM models on the ADHD200 dataset and explore the field of neuroscience. In this work, multiclass classification is performed on phenotypic data using an SVM model. The better results have been analyzed on the phenotypic data compared to other supervised learning techniques like Logistic regression, KNN, AdaBoost, etc. In addition, neural networks have been implemented on functional connectivity from the MRI data of a sample of 40 subjects provided to achieve high accuracy without prior knowledge of neuroscience. It is combined with the phenotypic classifier using the ensemble technique to get a binary classifier. It is further trained and tested on 400 out of 824 subjects from the ADHD200 data set and achieved an accuracy of 92.5% for binary classification The training and testing accuracy has been achieved upto 99% using ensemble classifier.

Sahil Goyal, Swayam Agrawal, Gautham Govind Anil et al.

We introduce Elastic Looped Transformers (ELT), a highly parameter-efficient class of visual generative models based on a recurrent transformer architecture. While conventional generative models rely on deep stacks of unique transformer layers, our approach employs iterative, weight-shared transformer blocks to drastically reduce parameter counts while maintaining high synthesis quality. To effectively train these models for image and video generation, we propose the idea of Intra-Loop Self Distillation (ILSD), where student configurations (intermediate loops) are distilled from the teacher configuration (maximum training loops) to ensure consistency across the model's depth in a single training step. Our framework yields a family of elastic models from a single training run, enabling Any-Time inference capability with dynamic trade-offs between computational cost and generation quality, with the same parameter count. ELT significantly shifts the efficiency frontier for visual synthesis. With $4\times$ reduction in parameter count under iso-inference-compute settings, ELT achieves a competitive FID of $2.0$ on class-conditional ImageNet $256 \times 256$ and FVD of $72.8$ on class-conditional UCF-101.

Abhishek Sharma, Pranjal Sharma, Darshan Pincha et al.

Nowadays, yoga has gained worldwide attention because of increasing levels of stress in the modern way of life, and there are many ways or resources to learn yoga. The word yoga means a deep connection between the mind and body. Today there is substantial Medical and scientific evidence to show that the very fundamentals of the activity of our brain, our chemistry even our genetic content can be changed by practicing different systems of yoga. Suryanamaskar, also known as salute to the sun, is a yoga practice that combines eight different forms and 12 asanas(4 asana get repeated) devoted to the Hindu Sun God, Surya. Suryanamaskar offers a number of health benefits such as strengthening muscles and helping to control blood sugar levels. Here the Mediapipe Library is used to analyze Surya namaskar situations. Standing is detected in real time with advanced software, as one performs Surya namaskar in front of the camera. The class divider identifies the form as one of the following: Pranamasana, Hasta Padasana, Hasta Uttanasana, Ashwa - Sanchalan asana, Ashtanga Namaskar, Dandasana, or Bhujangasana and Svanasana. Deep learning-based techniques(CNN) are used to develop this model with model accuracy of 98.68 percent and an accuracy score of 0.75 to detect correct yoga (Surya Namaskar ) posture. With this method, the users can practice the desired pose and can check if the pose that the person is doing is correct or not. It will help in doing all the different poses of surya namaskar correctly and increase the efficiency of the yoga practitioner. This paper describes the whole framework which is to be implemented in the model.

Oindrila Saha, Aditya Kusupati, Harsha Vardhan Simhadri et al.

Standard Convolutional Neural Networks (CNNs) designed for computer vision tasks tend to have large intermediate activation maps. These require large working memory and are thus unsuitable for deployment on resource-constrained devices typically used for inference on the edge. Aggressively downsampling the images via pooling or strided convolutions can address the problem but leads to a significant decrease in accuracy due to gross aggregation of the feature map by standard pooling operators. In this paper, we introduce RNNPool, a novel pooling operator based on Recurrent Neural Networks (RNNs), that efficiently aggregates features over large patches of an image and rapidly downsamples activation maps. Empirical evaluation indicates that an RNNPool layer can effectively replace multiple blocks in a variety of architectures such as MobileNets, DenseNet when applied to standard vision tasks like image classification and face detection. That is, RNNPool can significantly decrease computational complexity and peak memory usage for inference while retaining comparable accuracy. We use RNNPool with the standard S3FD architecture to construct a face detection method that achieves state-of-the-art MAP for tiny ARM Cortex-M4 class microcontrollers with under 256 KB of RAM. Code is released at https://github.com/Microsoft/EdgeML.

Ansh Makwe, Akansh Agrawal, Prateek Jain et al.

Medical image analysis for complex tasks such as severity grading and disease subtype classification poses significant challenges due to intricate and similar visual patterns among classes, scarcity of labeled data, and variability in expert interpretations. Despite the usefulness of existing attention-based models in capturing complex visual patterns for medical image classification, underlying architectures often face challenges in effectively distinguishing subtle classes since they struggle to capture inter-class similarity and intra-class variability, resulting in incorrect diagnosis. To address this, we propose AGGRNet framework to extract informative and non-informative features to effectively understand fine-grained visual patterns and improve classification for complex medical image analysis tasks. Experimental results show that our model achieves state-of-the-art performance on various medical imaging datasets, with the best improvement up to 5% over SOTA models on the Kvasir dataset.

Pranav Nair, Puranjay Datta, Jeff Dean et al. · uw

Quantizing model weights is critical for reducing the communication and inference costs of large models. However, quantizing models -- especially to low precisions like int4 or int2 -- requires a trade-off in model quality; int2, in particular, is known to severely degrade model quality. Consequently, practitioners are often forced to maintain multiple models with different quantization levels or serve a single model that best satisfies the quality-latency trade-off. On the other hand, integer data types, such as int8, inherently possess a nested (Matryoshka) structure where smaller bit-width integers, like int4 or int2, are nested within the most significant bits. Leveraging this insight, in this paper, we propose Matryoshka Quantization (MatQuant), a novel multi-scale quantization technique that alleviates the aforementioned challenge. This technique allows us to train and maintain a single quantized model but serve it with the precision demanded by the deployment. Furthermore, leveraging MatQuant's co-training and co-distillation regularization, int2 precision models extracted by MatQuant outperform standard int2 quantization by up to to 4% and 7% with OmniQuant and QAT as base algorithms respectively. Finally, we demonstrate that by using an extra bit to represent outliers, a model with an effective precision of 2.05-bit gives an additional 6% improvement with OmniQuant as the base algorithm.

Abhishek Sharma, Yash Shah, Yash Agrawal et al.

Nowadays, yoga has become a part of life for many people. Exercises and sports technological assistance is implemented in yoga pose identification. In this work, a self-assistance based yoga posture identification technique is developed, which helps users to perform Yoga with the correction feature in Real-time. The work also presents Yoga-hand mudra (hand gestures) identification. The YOGI dataset has been developed which include 10 Yoga postures with around 400-900 images of each pose and also contain 5 mudras for identification of mudras postures. It contains around 500 images of each mudra. The feature has been extracted by making a skeleton on the body for yoga poses and hand for mudra poses. Two different algorithms have been used for creating a skeleton one for yoga poses and the second for hand mudras. Angles of the joints have been extracted as a features for different machine learning and deep learning models. among all the models XGBoost with RandomSearch CV is most accurate and gives 99.2\% accuracy. The complete design framework is described in the present paper.

Naman Agarwal, Syomantak Chaudhuri, Prateek Jain et al.

Q-learning is a popular Reinforcement Learning (RL) algorithm which is widely used in practice with function approximation (Mnih et al., 2015). In contrast, existing theoretical results are pessimistic about Q-learning. For example, (Baird, 1995) shows that Q-learning does not converge even with linear function approximation for linear MDPs. Furthermore, even for tabular MDPs with synchronous updates, Q-learning was shown to have sub-optimal sample complexity (Li et al., 2021;Azar et al., 2013). The goal of this work is to bridge the gap between practical success of Q-learning and the relatively pessimistic theoretical results. The starting point of our work is the observation that in practice, Q-learning is used with two important modifications: (i) training with two networks, called online network and target network simultaneously (online target learning, or OTL) , and (ii) experience replay (ER) (Mnih et al., 2015). While they have been observed to play a significant role in the practical success of Q-learning, a thorough theoretical understanding of how these two modifications improve the convergence behavior of Q-learning has been missing in literature. By carefully combining Q-learning with OTL and reverse experience replay (RER) (a form of experience replay), we present novel methods Q-Rex and Q-RexDaRe (Q-Rex + data reuse). We show that Q-Rex efficiently finds the optimal policy for linear MDPs (or more generally for MDPs with zero inherent Bellman error with linear approximation (ZIBEL)) and provide non-asymptotic bounds on sample complexity -- the first such result for a Q-learning method for this class of MDPs under standard assumptions. Furthermore, we demonstrate that Q-RexDaRe in fact achieves near optimal sample complexity in the tabular setting, improving upon the existing results for vanilla Q-learning.

Kiran Koshy Thekumparampil, Prateek Jain, Praneeth Netrapalli et al.

We consider the classical setting of optimizing a nonsmooth Lipschitz continuous convex function over a convex constraint set, when having access to a (stochastic) first-order oracle (FO) for the function and a projection oracle (PO) for the constraint set. It is well known that to achieve $ε$-suboptimality in high-dimensions, $Θ(ε^{-2})$ FO calls are necessary. This is achieved by the projected subgradient method (PGD). However, PGD also entails $O(ε^{-2})$ PO calls, which may be computationally costlier than FO calls (e.g. nuclear norm constraints). Improving this PO calls complexity of PGD is largely unexplored, despite the fundamental nature of this problem and extensive literature. We present first such improvement. This only requires a mild assumption that the objective function, when extended to a slightly larger neighborhood of the constraint set, still remains Lipschitz and accessible via FO. In particular, we introduce MOPES method, which carefully combines Moreau-Yosida smoothing and accelerated first-order schemes. This is guaranteed to find a feasible $ε$-suboptimal solution using only $O(ε^{-1})$ PO calls and optimal $O(ε^{-2})$ FO calls. Further, instead of a PO if we only have a linear minimization oracle (LMO, a la Frank-Wolfe) to access the constraint set, an extension of our method, MOLES, finds a feasible $ε$-suboptimal solution using $O(ε^{-2})$ LMO calls and FO calls---both match known lower bounds, resolving a question left open since White (1993). Our experiments confirm that these methods achieve significant speedups over the state-of-the-art, for a problem with costly PO and LMO calls.

Bhaskar Mukhoty, Govind Gopakumar, Prateek Jain et al.

We provide the first global model recovery results for the IRLS (iteratively reweighted least squares) heuristic for robust regression problems. IRLS is known to offer excellent performance, despite bad initializations and data corruption, for several parameter estimation problems. Existing analyses of IRLS frequently require careful initialization, thus offering only local convergence guarantees. We remedy this by proposing augmentations to the basic IRLS routine that not only offer guaranteed global recovery, but in practice also outperform state-of-the-art algorithms for robust regression. Our routines are more immune to hyperparameter misspecification in basic regression tasks, as well as applied tasks such as linear-armed bandit problems. Our theoretical analyses rely on a novel extension of the notions of strong convexity and smoothness to weighted strong convexity and smoothness, and establishing that sub-Gaussian designs offer bounded weighted condition numbers. These notions may be useful in analyzing other algorithms as well.

Amar Budhiraja, Gaurush Hiranandani, Darshak Chhatbar et al.

In this paper, we study the problem of recommendation system where the users and items to be recommended are rich data structures with multiple entity types and with multiple sources of side-information in the form of graphs. We provide a general formulation for the problem that captures the complexities of modern real-world recommendations and generalizes many existing formulations. In our formulation, each user/document that requires a recommendation and each item or tag that is to be recommended, both are modeled by a set of static entities and a dynamic component. The relationships between entities are captured by several weighted bipartite graphs. To effectively exploit these complex interactions and learn the recommendation model, we propose MEDRES- a multiple graph-CNN based novel deep-learning architecture. MEDRES uses AL-GCN, a novel graph convolution network block, that harnesses strong representative features from the underlying graphs. Moreover, in order to capture highly heterogeneous engagement of different users with the system and constraints on the number of items to be recommended, we propose a novel ranking metric pAp@k along with a method to optimize the metric directly. We demonstrate effectiveness of our method on two benchmarks: a) citation data, b) Flickr data. In addition, we present two real-world case studies of our formulation and the MEDRES architecture. We show how our technique can be used to naturally model the message recommendation problem and the teams recommendation problem in the Microsoft Teams (MSTeams) product and demonstrate that it is 5-6% points more accurate than the production-grade models.

Kiran Koshy Thekumparampil, Prateek Jain, Praneeth Netrapalli et al.

This paper studies first order methods for solving smooth minimax optimization problems $\min_x \max_y g(x,y)$ where $g(\cdot,\cdot)$ is smooth and $g(x,\cdot)$ is concave for each $x$. In terms of $g(\cdot,y)$, we consider two settings -- strongly convex and nonconvex -- and improve upon the best known rates in both. For strongly-convex $g(\cdot, y),\ \forall y$, we propose a new algorithm combining Mirror-Prox and Nesterov's AGD, and show that it can find global optimum in $\tilde{O}(1/k^2)$ iterations, improving over current state-of-the-art rate of $O(1/k)$. We use this result along with an inexact proximal point method to provide $\tilde{O}(1/k^{1/3})$ rate for finding stationary points in the nonconvex setting where $g(\cdot, y)$ can be nonconvex. This improves over current best-known rate of $O(1/k^{1/5})$. Finally, we instantiate our result for finite nonconvex minimax problems, i.e., $\min_x \max_{1\leq i\leq m} f_i(x)$, with nonconvex $f_i(\cdot)$, to obtain convergence rate of $O(m(\log m)^{3/2}/k^{1/3})$ total gradient evaluations for finding a stationary point.

Arun Sai Suggala, Kush Bhatia, Pradeep Ravikumar et al.

We study the problem of robust linear regression with response variable corruptions. We consider the oblivious adversary model, where the adversary corrupts a fraction of the responses in complete ignorance of the data. We provide a nearly linear time estimator which consistently estimates the true regression vector, even with $1-o(1)$ fraction of corruptions. Existing results in this setting either don't guarantee consistent estimates or can only handle a small fraction of corruptions. We also extend our estimator to robust sparse linear regression and show that similar guarantees hold in this setting. Finally, we apply our estimator to the problem of linear regression with heavy-tailed noise and show that our estimator consistently estimates the regression vector even when the noise has unbounded variance (e.g., Cauchy distribution), for which most existing results don't even apply. Our estimator is based on a novel variant of outlier removal via hard thresholding in which the threshold is chosen adaptively and crucially relies on randomness to escape bad fixed points of the non-convex hard thresholding operation.

Kai Zhong, Zhao Song, Prateek Jain et al.

The goal of a recommendation system is to predict the interest of a user in a given item by exploiting the existing set of ratings as well as certain user/item features. A standard approach to modeling this problem is Inductive Matrix Completion where the predicted rating is modeled as an inner product of the user and the item features projected onto a latent space. In order to learn the parameters effectively from a small number of observed ratings, the latent space is constrained to be low-dimensional which implies that the parameter matrix is constrained to be low-rank. However, such bilinear modeling of the ratings can be limiting in practice and non-linear prediction functions can lead to significant improvements. A natural approach to introducing non-linearity in the prediction function is to apply a non-linear activation function on top of the projected user/item features. Imposition of non-linearities further complicates an already challenging problem that has two sources of non-convexity: a) low-rank structure of the parameter matrix, and b) non-linear activation function. We show that one can still solve the non-linear Inductive Matrix Completion problem using gradient descent type methods as long as the solution is initialized well. That is, close to the optima, the optimization function is strongly convex and hence admits standard optimization techniques, at least for certain activation functions, such as Sigmoid and tanh. We also highlight the importance of the activation function and show how ReLU can behave significantly differently than say a sigmoid function. Finally, we apply our proposed technique to recommendation systems and semi-supervised clustering, and show that our method can lead to much better performance than standard linear Inductive Matrix Completion methods.

Srinadh Bhojanapalli, Nicolas Boumal, Prateek Jain et al.

Semidefinite programs (SDP) are important in learning and combinatorial optimization with numerous applications. In pursuit of low-rank solutions and low complexity algorithms, we consider the Burer--Monteiro factorization approach for solving SDPs. We show that all approximate local optima are global optima for the penalty formulation of appropriately rank-constrained SDPs as long as the number of constraints scales sub-quadratically with the desired rank of the optimal solution. Our result is based on a simple penalty function formulation of the rank-constrained SDP along with a smoothed analysis to avoid worst-case cost matrices. We particularize our results to two applications, namely, Max-Cut and matrix completion.

Yeshwanth Cherapanamjeri, Prateek Jain, Praneeth Netrapalli

We consider the problem of outlier robust PCA (OR-PCA) where the goal is to recover principal directions despite the presence of outlier data points. That is, given a data matrix $M^*$, where $(1-α)$ fraction of the points are noisy samples from a low-dimensional subspace while $α$ fraction of the points can be arbitrary outliers, the goal is to recover the subspace accurately. Existing results for \OR-PCA have serious drawbacks: while some results are quite weak in the presence of noise, other results have runtime quadratic in dimension, rendering them impractical for large scale applications. In this work, we provide a novel thresholding based iterative algorithm with per-iteration complexity at most linear in the data size. Moreover, the fraction of outliers, $α$, that our method can handle is tight up to constants while providing nearly optimal computational complexity for a general noise setting. For the special case where the inliers are obtained from a low-dimensional subspace with additive Gaussian noise, we show that a modification of our thresholding based method leads to significant improvement in recovery error (of the subspace) even in the presence of a large fraction of outliers.

Kai Zhong, Prateek Jain, Ashish Kapoor

This paper considers the problem of safe mission planning of dynamic systems operating under uncertain environments. Much of the prior work on achieving robust and safe control requires solving second-order cone programs (SOCP). Unfortunately, existing general purpose SOCP methods are often infeasible for real-time robotic tasks due to high memory and computational requirements imposed by existing general optimization methods. The key contribution of this paper is a fast and memory-efficient algorithm for SOCP that would enable robust and safe mission planning on-board robots in real-time. Our algorithm does not have any external dependency, can efficiently utilize warm start provided in safe planning settings, and in fact leads to significant speed up over standard optimization packages (like SDPT3) for even standard SOCP problems. For example, for a standard quadrotor problem, our method leads to speedup of 1000x over SDPT3 without any deterioration in the solution quality. Our method is based on two insights: a) SOCPs can be interpreted as optimizing a function over a polytope with infinite sides, b) a linear function can be efficiently optimized over this polytope. We combine the above observations with a novel utilization of Wolfe's algorithm to obtain an efficient optimization method that can be easily implemented on small embedded devices. In addition to the above mentioned algorithm, we also design a two-level sensing method based on Gaussian Process for complex obstacles with non-linear boundaries such as a cylinder.

Kush Bhatia, Prateek Jain, Parameswaran Kamalaruban et al.

We study the problem of robust time series analysis under the standard auto-regressive (AR) time series model in the presence of arbitrary outliers. We devise an efficient hard thresholding based algorithm which can obtain a consistent estimate of the optimal AR model despite a large fraction of the time series points being corrupted. Our algorithm alternately estimates the corrupted set of points and the model parameters, and is inspired by recent advances in robust regression and hard-thresholding methods. However, a direct application of existing techniques is hindered by a critical difference in the time-series domain: each point is correlated with all previous points rendering existing tools inapplicable directly. We show how to overcome this hurdle using novel proof techniques. Using our techniques, we are also able to provide the first efficient and provably consistent estimator for the robust regression problem where a standard linear observation model with white additive noise is corrupted arbitrarily. We illustrate our methods on synthetic datasets and show that our methods indeed are able to consistently recover the optimal parameters despite a large fraction of points being corrupted.

Yeshwanth Cherapanamjeri, Kartik Gupta, Prateek Jain

In this paper, we consider the problem of Robust Matrix Completion (RMC) where the goal is to recover a low-rank matrix by observing a small number of its entries out of which a few can be arbitrarily corrupted. We propose a simple projected gradient descent method to estimate the low-rank matrix that alternately performs a projected gradient descent step and cleans up a few of the corrupted entries using hard-thresholding. Our algorithm solves RMC using nearly optimal number of observations as well as nearly optimal number of corruptions. Our result also implies significant improvement over the existing time complexity bounds for the low-rank matrix completion problem. Finally, an application of our result to the robust PCA problem (low-rank+sparse matrix separation) leads to nearly linear time (in matrix dimensions) algorithm for the same; existing state-of-the-art methods require quadratic time. Our empirical results corroborate our theoretical results and show that even for moderate sized problems, our method for robust PCA is an an order of magnitude faster than the existing methods.

Prateek Jain, Chi Jin, Sham M. Kakade et al.

This work provides improved guarantees for streaming principle component analysis (PCA). Given $A_1, \ldots, A_n\in \mathbb{R}^{d\times d}$ sampled independently from distributions satisfying $\mathbb{E}[A_i] = Σ$ for $Σ\succeq \mathbf{0}$, this work provides an $O(d)$-space linear-time single-pass streaming algorithm for estimating the top eigenvector of $Σ$. The algorithm nearly matches (and in certain cases improves upon) the accuracy obtained by the standard batch method that computes top eigenvector of the empirical covariance $\frac{1}{n} \sum_{i \in [n]} A_i$ as analyzed by the matrix Bernstein inequality. Moreover, to achieve constant accuracy, our algorithm improves upon the best previous known sample complexities of streaming algorithms by either a multiplicative factor of $O(d)$ or $1/\mathrm{gap}$ where $\mathrm{gap}$ is the relative distance between the top two eigenvalues of $Σ$. These results are achieved through a novel analysis of the classic Oja's algorithm, one of the oldest and most popular algorithms for streaming PCA. In particular, this work shows that simply picking a random initial point $w_0$ and applying the update rule $w_{i + 1} = w_i + η_i A_i w_i$ suffices to accurately estimate the top eigenvector, with a suitable choice of $η_i$. We believe our result sheds light on how to efficiently perform streaming PCA both in theory and in practice and we hope that our analysis may serve as the basis for analyzing many variants and extensions of streaming PCA.

Prateek Jain, Nikhil Rao, Inderjit Dhillon

Several learning applications require solving high-dimensional regression problems where the relevant features belong to a small number of (overlapping) groups. For very large datasets and under standard sparsity constraints, hard thresholding methods have proven to be extremely efficient, but such methods require NP hard projections when dealing with overlapping groups. In this paper, we show that such NP-hard projections can not only be avoided by appealing to submodular optimization, but such methods come with strong theoretical guarantees even in the presence of poorly conditioned data (i.e. say when two features have correlation $\geq 0.99$), which existing analyses cannot handle. These methods exhibit an interesting computation-accuracy trade-off and can be extended to significantly harder problems such as sparse overlapping groups. Experiments on both real and synthetic data validate our claims and demonstrate that the proposed methods are orders of magnitude faster than other greedy and convex relaxation techniques for learning with group-structured sparsity.

Animashree Anandkumar, Prateek Jain, Yang Shi et al.

Robust tensor CP decomposition involves decomposing a tensor into low rank and sparse components. We propose a novel non-convex iterative algorithm with guaranteed recovery. It alternates between low-rank CP decomposition through gradient ascent (a variant of the tensor power method), and hard thresholding of the residual. We prove convergence to the globally optimal solution under natural incoherence conditions on the low rank component, and bounded level of sparse perturbations. We compare our method with natural baselines which apply robust matrix PCA either to the {\em flattened} tensor, or to the matrix slices of the tensor. Our method can provably handle a far greater level of perturbation when the sparse tensor is block-structured. This naturally occurs in many applications such as the activity detection task in videos. Our experiments validate these findings. Thus, we establish that tensor methods can tolerate a higher level of gross corruptions compared to matrix methods.

Vidyadhar Rao, Prateek Jain, C. V Jawahar

Typical retrieval systems have three requirements: a) Accurate retrieval i.e., the method should have high precision, b) Diverse retrieval, i.e., the obtained set of points should be diverse, c) Retrieval time should be small. However, most of the existing methods address only one or two of the above mentioned requirements. In this work, we present a method based on randomized locality sensitive hashing which tries to address all of the above requirements simultaneously. While earlier hashing approaches considered approximate retrieval to be acceptable only for the sake of efficiency, we argue that one can further exploit approximate retrieval to provide impressive trade-offs between accuracy and diversity. We extend our method to the problem of multi-label prediction, where the goal is to output a diverse and accurate set of labels for a given document in real-time. Moreover, we introduce a new notion to simultaneously evaluate a method's performance for both the precision and diversity measures. Finally, we present empirical results on several different retrieval tasks and show that our method retrieves diverse and accurate images/labels while ensuring $100x$-speed-up over the existing diverse retrieval approaches.

Kush Bhatia, Prateek Jain, Purushottam Kar

We study the problem of Robust Least Squares Regression (RLSR) where several response variables can be adversarially corrupted. More specifically, for a data matrix X \in R^{p x n} and an underlying model w*, the response vector is generated as y = X'w* + b where b \in R^n is the corruption vector supported over at most C.n coordinates. Existing exact recovery results for RLSR focus solely on L1-penalty based convex formulations and impose relatively strict model assumptions such as requiring the corruptions b to be selected independently of X. In this work, we study a simple hard-thresholding algorithm called TORRENT which, under mild conditions on X, can recover w* exactly even if b corrupts the response variables in an adversarial manner, i.e. both the support and entries of b are selected adversarially after observing X and w*. Our results hold under deterministic assumptions which are satisfied if X is sampled from any sub-Gaussian distribution. Finally unlike existing results that apply only to a fixed w*, generated independently of X, our results are universal and hold for any w* \in R^p. Next, we propose gradient descent-based extensions of TORRENT that can scale efficiently to large scale problems, such as high dimensional sparse recovery and prove similar recovery guarantees for these extensions. Empirically we find TORRENT, and more so its extensions, offering significantly faster recovery than the state-of-the-art L1 solvers. For instance, even on moderate-sized datasets (with p = 50K) with around 40% corrupted responses, a variant of our proposed method called TORRENT-HYB is more than 20x faster than the best L1 solver.

Prateek Jain, Ambuj Tewari, Purushottam Kar

The use of M-estimators in generalized linear regression models in high dimensional settings requires risk minimization with hard $L_0$ constraints. Of the known methods, the class of projected gradient descent (also known as iterative hard thresholding (IHT)) methods is known to offer the fastest and most scalable solutions. However, the current state-of-the-art is only able to analyze these methods in extremely restrictive settings which do not hold in high dimensional statistical models. In this work we bridge this gap by providing the first analysis for IHT-style methods in the high dimensional statistical setting. Our bounds are tight and match known minimax lower bounds. Our results rely on a general analysis framework that enables us to analyze several popular hard thresholding style algorithms (such as HTP, CoSaMP, SP) in the high dimensional regression setting. We also extend our analysis to a large family of "fully corrective methods" that includes two-stage and partial hard-thresholding algorithms. We show that our results hold for the problem of sparse regression, as well as low-rank matrix recovery.

Prateek Jain, Sewoong Oh

We study the problem of low-rank tensor factorization in the presence of missing data. We ask the following question: how many sampled entries do we need, to efficiently and exactly reconstruct a tensor with a low-rank orthogonal decomposition? We propose a novel alternating minimization based method which iteratively refines estimates of the singular vectors. We show that under certain standard assumptions, our method can recover a three-mode $n\times n\times n$ dimensional rank-$r$ tensor exactly from $O(n^{3/2} r^5 \log^4 n)$ randomly sampled entries. In the process of proving this result, we solve two challenging sub-problems for tensors with missing data. First, in the process of analyzing the initialization step, we prove a generalization of a celebrated result by Szemerédie et al. on the spectrum of random graphs. Next, we prove global convergence of alternating minimization with a good initialization. Simulations suggest that the dependence of the sample size on dimensionality $n$ is indeed tight.