arXiv subjects

Rene Vidal

Publications and source records attributed to Rene Vidal.

At least 37 records · Page 2Linked to original sources

arXiv2023

Variational Information Pursuit with Large Language and Multimodal Models for Interpretable Predictions

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.

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arXiv2023

Learning Globally Smooth Functions on Manifolds

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.

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arXiv2022

Interpretable by Design: Learning Predictors by Composing Interpretable Queries

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.

cs.CV↗

arXiv2022

Structured Graph Variational Autoencoders for Indoor Furniture layout Generation

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.

cs.CV↗

arXiv2022

Convergence and Implicit Bias of Gradient Flow on Overparametrized Linear Networks

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.

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arXiv2022

The Vision of Self-Evolving Computing Systems

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.

cs.SE↗

arXiv2021

Boosting RANSAC via Dual Principal Component Pursuit

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.

cs.CV↗

arXiv2020

What is the Largest Sparsity Pattern that Can Be Recovered by 1-Norm Minimization?

Much of the existing literature in sparse recovery is concerned with the following question: given a sparsity pattern and a corresponding regularizer, derive conditions on the dictionary under which exact recovery is possible. In this paper, we study the opposite question: given a dictionary and the 1-norm regularizer, find the largest sparsity pattern that can be recovered. We show that such a pattern is described by a mathematical object called a "maximum abstract simplicial complex", and provide two different characterizations of this object: one based on extreme points and the other based on vectors of minimal support. In addition, we show how this new framework is useful in the study of sparse recovery problems when the dictionary takes the form of a graph incidence matrix or a partial discrete Fourier transform. In case of incidence matrices, we show that the largest sparsity pattern that can be recovered is determined by the set of simple cycles of the graph. As a byproduct, we show that standard sparse recovery can be certified in polynomial time, although this is known to be NP-hard for general matrices. In the case of the partial discrete Fourier transform, our characterization of the largest sparsity pattern that can be recovered requires the unknown signal to be real and its dimension to be a prime number.

eess.SP↗

arXiv2020

Self-Representation Based Unsupervised Exemplar Selection in a Union of Subspaces

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.

cs.LG↗

arXiv2020

Is an Affine Constraint Needed for Affine Subspace Clustering?

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.

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arXiv2019

Basis Pursuit and Orthogonal Matching Pursuit for Subspace-preserving Recovery: Theoretical Analysis

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.

cs.LG↗

arXiv2019

Dual Principal Component Pursuit

We consider the problem of learning a linear subspace from data corrupted by outliers. Classical approaches are typically designed for the case in which the subspace dimension is small relative to the ambient dimension. Our approach works with a dual representation of the subspace and hence aims to find its orthogonal complement; as such, it is particularly suitable for subspaces whose dimension is close to the ambient dimension (subspaces of high relative dimension). We pose the problem of computing normal vectors to the inlier subspace as a non-convex $\ell_1$ minimization problem on the sphere, which we call Dual Principal Component Pursuit (DPCP) problem. We provide theoretical guarantees under which every global solution to DPCP is a vector in the orthogonal complement of the inlier subspace. Moreover, we relax the non-convex DPCP problem to a recursion of linear programs whose solutions are shown to converge in a finite number of steps to a vector orthogonal to the subspace. In particular, when the inlier subspace is a hyperplane, the solutions to the recursion of linear programs converge to the global minimum of the non-convex DPCP problem in a finite number of steps. We also propose algorithms based on alternating minimization and iteratively re-weighted least squares, which are suitable for dealing with large-scale data. Experiments on synthetic data show that the proposed methods are able to handle more outliers and higher relative dimensions than current state-of-the-art methods, while experiments in the context of the three-view geometry problem in computer vision suggest that the proposed methods can be a useful or even superior alternative to traditional RANSAC-based approaches for computer vision and other applications.

cs.CV↗

arXiv2019

Nonconvex Robust Low-rank Matrix Recovery

In this paper we study the problem of recovering a low-rank matrix from a number of random linear measurements that are corrupted by outliers taking arbitrary values. We consider a nonsmooth nonconvex formulation of the problem, in which we explicitly enforce the low-rank property of the solution by using a factored representation of the matrix variable and employ an $\ell_1$-loss function to robustify the solution against outliers. We show that even when a constant fraction (which can be up to almost half) of the measurements are arbitrarily corrupted, as long as certain measurement operators arising from the measurement model satisfy the so-called $\ell_1/\ell_2$-restricted isometry property, the ground-truth matrix can be exactly recovered from any global minimum of the resulting optimization problem. Furthermore, we show that the objective function of the optimization problem is sharp and weakly convex. Consequently, a subgradient Method (SubGM) with geometrically diminishing step sizes will converge linearly to the ground-truth matrix when suitably initialized. We demonstrate the efficacy of the SubGM for the nonconvex robust low-rank matrix recovery problem with various numerical experiments.

cs.IT↗

arXiv2018

Dual Principal Component Pursuit: Probability Analysis and Efficient Algorithms

Recent methods for learning a linear subspace from data corrupted by outliers are based on convex $\ell_1$ and nuclear norm optimization and require the dimension of the subspace and the number of outliers to be sufficiently small. In sharp contrast, the recently proposed Dual Principal Component Pursuit (DPCP) method can provably handle subspaces of high dimension by solving a non-convex $\ell_1$ optimization problem on the sphere. However, its geometric analysis is based on quantities that are difficult to interpret and are not amenable to statistical analysis. In this paper we provide a refined geometric analysis and a new statistical analysis that show that DPCP can tolerate as many outliers as the square of the number of inliers, thus improving upon other provably correct robust PCA methods. We also propose a scalable Projected Sub-Gradient Method method (DPCP-PSGM) for solving the DPCP problem and show it admits linear convergence even though the underlying optimization problem is non-convex and non-smooth. Experiments on road plane detection from 3D point cloud data demonstrate that DPCP-PSGM can be more efficient than the traditional RANSAC algorithm, which is one of the most popular methods for such computer vision applications.

cs.LG↗

arXiv2018

Stretching Domain Adaptation: How far is too far?

While deep learning has led to significant advances in visual recognition over the past few years, such advances often require a lot of annotated data. Unsupervised domain adaptation has emerged as an alternative approach that does not require as much annotated data, prior evaluations of domain adaptation approaches have been limited to relatively similar datasets, e.g source and target domains are samples captured by different cameras. A new data suite is proposed that comprehensively evaluates cross-modality domain adaptation problems. This work pushes the limit of unsupervised domain adaptation through an in-depth evaluation of several state of the art methods on benchmark datasets and the new dataset suite. We also propose a new domain adaptation network called "Deep MagNet" that effectively transfers knowledge for cross-modality domain adaptation problems. Deep Magnet achieves state of the art performance on two benchmark datasets. More importantly, the proposed method shows consistent improvements in performance on the newly proposed dataset suite.

cs.CV↗

arXiv2018

Convolutional Networks for Object Category and 3D Pose Estimation from 2D Images

Current CNN-based algorithms for recovering the 3D pose of an object in an image assume knowledge about both the object category and its 2D localization in the image. In this paper, we relax one of these constraints and propose to solve the task of joint object category and 3D pose estimation from an image assuming known 2D localization. We design a new architecture for this task composed of a feature network that is shared between subtasks, an object categorization network built on top of the feature network, and a collection of category dependent pose regression networks. We also introduce suitable loss functions and a training method for the new architecture. Experiments on the challenging PASCAL3D+ dataset show state-of-the-art performance in the joint categorization and pose estimation task. Moreover, our performance on the joint task is comparable to the performance of state-of-the-art methods on the simpler 3D pose estimation with known object category task.

cs.CV↗

arXiv2018

On the Implicit Bias of Dropout

Algorithmic approaches endow deep learning systems with implicit bias that helps them generalize even in over-parametrized settings. In this paper, we focus on understanding such a bias induced in learning through dropout, a popular technique to avoid overfitting in deep learning. For single hidden-layer linear neural networks, we show that dropout tends to make the norm of incoming/outgoing weight vectors of all the hidden nodes equal. In addition, we provide a complete characterization of the optimization landscape induced by dropout.

cs.LG↗

arXiv2018

A Mixed Classification-Regression Framework for 3D Pose Estimation from 2D Images

3D pose estimation from a single 2D image is an important and challenging task in computer vision with applications in autonomous driving, robot manipulation and augmented reality. Since 3D pose is a continuous quantity, a natural formulation for this task is to solve a pose regression problem. However, since pose regression methods return a single estimate of the pose, they have difficulties handling multimodal pose distributions (e.g. in the case of symmetric objects). An alternative formulation, which can capture multimodal pose distributions, is to discretize the pose space into bins and solve a pose classification problem. However, pose classification methods can give large pose estimation errors depending on the coarseness of the discretization. In this paper, we propose a mixed classification-regression framework that uses a classification network to produce a discrete multimodal pose estimate and a regression network to produce a continuous refinement of the discrete estimate. The proposed framework can accommodate different architectures and loss functions, leading to multiple classification-regression models, some of which achieve state-of-the-art performance on the challenging Pascal3D+ dataset.

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