arXiv subjects

Gilad Lerman

Publications and source records attributed to Gilad Lerman.

At least 37 records · Page 2Linked to original sources

arXiv2024

Monotone Generative Modeling via a Gromov-Monge Embedding

Generative adversarial networks (GANs) are popular for generative tasks; however, they often require careful architecture selection, extensive empirical tuning, and are prone to mode collapse. To overcome these challenges, we propose a novel model that identifies the low-dimensional structure of the underlying data distribution, maps it into a low-dimensional latent space while preserving the underlying geometry, and then optimally transports a reference measure to the embedded distribution. We prove three key properties of our method: 1) The encoder preserves the geometry of the underlying data; 2) The generator is $c$-cyclically monotone, where $c$ is an intrinsic embedding cost employed by the encoder; and 3) The discriminator's modulus of continuity improves with the geometric preservation of the data. Numerical experiments demonstrate the effectiveness of our approach in generating high-quality images and exhibiting robustness to both mode collapse and training instability.

cs.LG↗

arXiv2024

A Subspace-Constrained Tyler's Estimator and its Applications to Structure from Motion

We present the subspace-constrained Tyler's estimator (STE) designed for recovering a low-dimensional subspace within a dataset that may be highly corrupted with outliers. STE is a fusion of the Tyler's M-estimator (TME) and a variant of the fast median subspace. Our theoretical analysis suggests that, under a common inlier-outlier model, STE can effectively recover the underlying subspace, even when it contains a smaller fraction of inliers relative to other methods in the field of robust subspace recovery. We apply STE in the context of Structure from Motion (SfM) in two ways: for robust estimation of the fundamental matrix and for the removal of outlying cameras, enhancing the robustness of the SfM pipeline. Numerical experiments confirm the state-of-the-art performance of our method in these applications. This research makes significant contributions to the field of robust subspace recovery, particularly in the context of computer vision and 3D reconstruction.

cs.CV↗

arXiv2024

The effect of Leaky ReLUs on the training and generalization of overparameterized networks

We investigate the training and generalization errors of overparameterized neural networks (NNs) with a wide class of leaky rectified linear unit (ReLU) functions. More specifically, we carefully upper bound both the convergence rate of the training error and the generalization error of such NNs and investigate the dependence of these bounds on the Leaky ReLU parameter, $α$. We show that $α=-1$, which corresponds to the absolute value activation function, is optimal for the training error bound. Furthermore, in special settings, it is also optimal for the generalization error bound. Numerical experiments empirically support the practical choices guided by the theory.

cs.LG↗

arXiv2023

Spectrally Constrained Optimization

We investigate how to solve smooth matrix optimization problems with general linear inequality constraints on the eigenvalues of a symmetric matrix. We present solution methods to obtain exact global minima for linear objective functions, i.e., $F(X) = \langle C, X \rangle$, and perform exact projections onto the eigenvalue constraint set. Two first-order algorithms are developed to obtain first-order stationary points for general non-convex objective functions. Both methods are proven to converge sublinearly when the constraint set is convex. Numerical experiments demonstrate the applicability of both the model and the methods.

math.OC↗

arXiv2023

An Unpooling Layer for Graph Generation

We propose a novel and trainable graph unpooling layer for effective graph generation. Given a graph with features, the unpooling layer enlarges this graph and learns its desired new structure and features. Since this unpooling layer is trainable, it can be applied to graph generation either in the decoder of a variational autoencoder or in the generator of a generative adversarial network (GAN). We prove that the unpooled graph remains connected and any connected graph can be sequentially unpooled from a 3-nodes graph. We apply the unpooling layer within the GAN generator. Since the most studied instance of graph generation is molecular generation, we test our ideas in this context. Using the QM9 and ZINC datasets, we demonstrate the improvement obtained by using the unpooling layer instead of an adjacency-matrix-based approach.

cs.LG↗

arXiv2023

Novelty Detection via Robust Variational Autoencoding

We propose a new method for novelty detection that can tolerate high corruption of the training points, whereas previous works assumed either no or very low corruption. Our method trains a robust variational autoencoder (VAE), which aims to generate a model for the uncorrupted training points. To gain robustness to high corruption, we incorporate the following four changes to the common VAE: 1. Extracting crucial features of the latent code by a carefully designed dimension reduction component for distributions; 2. Modeling the latent distribution as a mixture of Gaussian low-rank inliers and full-rank outliers, where the testing only uses the inlier model; 3. Applying the Wasserstein-1 metric for regularization, instead of the Kullback-Leibler (KL) divergence; and 4. Using a robust error for reconstruction. We establish both robustness to outliers and suitability to low-rank modeling of the Wasserstein metric as opposed to the KL divergence. We illustrate state-of-the-art results on standard benchmarks.

cs.LG↗

arXiv2022

Robust Vector Quantized-Variational Autoencoder

Image generative models can learn the distributions of the training data and consequently generate examples by sampling from these distributions. However, when the training dataset is corrupted with outliers, generative models will likely produce examples that are also similar to the outliers. In fact, a small portion of outliers may induce state-of-the-art generative models, such as Vector Quantized-Variational AutoEncoder (VQ-VAE), to learn a significant mode from the outliers. To mitigate this problem, we propose a robust generative model based on VQ-VAE, which we name Robust VQ-VAE (RVQ-VAE). In order to achieve robustness, RVQ-VAE uses two separate codebooks for the inliers and outliers. To ensure the codebooks embed the correct components, we iteratively update the sets of inliers and outliers during each training epoch. To ensure that the encoded data points are matched to the correct codebooks, we quantize using a weighted Euclidean distance, whose weights are determined by directional variances of the codebooks. Both codebooks, together with the encoder and decoder, are trained jointly according to the reconstruction loss and the quantization loss. We experimentally demonstrate that RVQ-VAE is able to generate examples from inliers even if a large portion of the training data points are corrupted.

cs.LG↗

arXiv2022

Cubic-Regularized Newton for Spectral Constrained Matrix Optimization and its Application to Fairness

Matrix functions are utilized to rewrite smooth spectral constrained matrix optimization problems as smooth unconstrained problems over the set of symmetric matrices which are then solved via the cubic-regularized Newton method. A second-order chain rule identity for matrix functions is proven to compute the higher-order derivatives to implement cubic-regularized Newton, and a new convergence analysis is provided for cubic-regularized Newton for matrix vector spaces. We demonstrate the applicability of our approach by conducting numerical experiments on both synthetic and real datasets. In our experiments, we formulate a new model for estimating fair and robust covariance matrices in the spirit of the Tyler's M-estimator (TME) model and demonstrate its advantage.

math.OC↗

arXiv2022

Robust Group Synchronization via Quadratic Programming

We propose a novel quadratic programming formulation for estimating the corruption levels in group synchronization, and use these estimates to solve this problem. Our objective function exploits the cycle consistency of the group and we thus refer to our method as detection and estimation of structural consistency (DESC). This general framework can be extended to other algebraic and geometric structures. Our formulation has the following advantages: it can tolerate corruption as high as the information-theoretic bound, it does not require a good initialization for the estimates of group elements, it has a simple interpretation, and under some mild conditions the global minimum of our objective function exactly recovers the corruption levels. We demonstrate the competitive accuracy of our approach on both synthetic and real data experiments of rotation averaging.

stat.ML↗

arXiv2022

Fast, Accurate and Memory-Efficient Partial Permutation Synchronization

Previous partial permutation synchronization (PPS) algorithms, which are commonly used for multi-object matching, often involve computation-intensive and memory-demanding matrix operations. These operations become intractable for large scale structure-from-motion datasets. For pure permutation synchronization, the recent Cycle-Edge Message Passing (CEMP) framework suggests a memory-efficient and fast solution. Here we overcome the restriction of CEMP to compact groups and propose an improved algorithm, CEMP-Partial, for estimating the corruption levels of the observed partial permutations. It allows us to subsequently implement a nonconvex weighted projected power method without the need of spectral initialization. The resulting new PPS algorithm, MatchFAME (Fast, Accurate and Memory-Efficient Matching), only involves sparse matrix operations, and thus enjoys lower time and space complexities in comparison to previous PPS algorithms. We prove that under adversarial corruption, though without additive noise and with certain assumptions, CEMP-Partial is able to exactly classify corrupted and clean partial permutations. We demonstrate the state-of-the-art accuracy, speed and memory efficiency of our method on both synthetic and real datasets.

cs.CV↗

arXiv2022

Depth Descent Synchronization in $\mathrm{SO}(D)$

We give robust recovery results for synchronization on the rotation group, $\mathrm{SO}(D)$. In particular, we consider an adversarial corruption setting, where a limited percentage of the observations are arbitrarily corrupted. We give a novel algorithm that exploits Tukey depth in the tangent space, which exactly recovers the underlying rotations up to an outlier percentage of $1/(D(D-1)+2)$. This corresponds to an outlier fraction of $1/4$ for $\mathrm{SO}(2)$ and $1/8$ for $\mathrm{SO}(3)$. In the case of $D=2$, we demonstrate that a variant of this algorithm converges linearly to the ground truth rotations. We finish by discussing this result in relation to a simpler nonconvex energy minimization framework based on least absolute deviations, which exhibits spurious fixed points.

math.OC↗

arXiv2022

Stochastic and Private Nonconvex Outlier-Robust PCA

We develop theoretically guaranteed stochastic methods for outlier-robust PCA. Outlier-robust PCA seeks an underlying low-dimensional linear subspace from a dataset that is corrupted with outliers. We are able to show that our methods, which involve stochastic geodesic gradient descent over the Grassmannian manifold, converge and recover an underlying subspace in various regimes through the development of a novel convergence analysis. The main application of this method is an effective differentially private algorithm for outlier-robust PCA that uses a Gaussian noise mechanism within the stochastic gradient method. Our results emphasize the advantages of the nonconvex methods over another convex approach to solving this problem in the differentially private setting. Experiments on synthetic and stylized data verify these results.

cs.LG↗

arXiv2022

Analysis and algorithms for $\ell_p$-based semi-supervised learning on graphs

This paper addresses theory and applications of $\ell_p$-based Laplacian regularization in semi-supervised learning. The graph $p$-Laplacian for $p>2$ has been proposed recently as a replacement for the standard ($p=2$) graph Laplacian in semi-supervised learning problems with very few labels, where Laplacian learning is degenerate. In the first part of the paper we prove new discrete to continuum convergence results for $p$-Laplace problems on $k$-nearest neighbor ($k$-NN) graphs, which are more commonly used in practice than random geometric graphs. Our analysis shows that, on $k$-NN graphs, the $p$-Laplacian retains information about the data distribution as $p\to \infty$ and Lipschitz learning ($p=\infty$) is sensitive to the data distribution. This situation can be contrasted with random geometric graphs, where the $p$-Laplacian forgets the data distribution as $p\to \infty$. We also present a general framework for proving discrete to continuum convergence results in graph-based learning that only requires pointwise consistency and monotonicity. In the second part of the paper, we develop fast algorithms for solving the variational and game-theoretic $p$-Laplace equations on weighted graphs for $p>2$. We present several efficient and scalable algorithms for both formulations, and present numerical results on synthetic data indicating their convergence properties. Finally, we conduct extensive numerical experiments on the MNIST, FashionMNIST and EMNIST datasets that illustrate the effectiveness of the $p$-Laplacian formulation for semi-supervised learning with few labels. In particular, we find that Lipschitz learning ($p=\infty$) performs well with very few labels on $k$-NN graphs, which experimentally validates our theoretical findings that Lipschitz learning retains information about the data distribution (the unlabeled data) on $k$-NN graphs.

math.NA↗

arXiv2022

Scalable Cluster-Consistency Statistics for Robust Multi-Object Matching

We develop new statistics for robustly filtering corrupted keypoint matches in the structure from motion pipeline. The statistics are based on consistency constraints that arise within the clustered structure of the graph of keypoint matches. The statistics are designed to give smaller values to corrupted matches and than uncorrupted matches. These new statistics are combined with an iterative reweighting scheme to filter keypoints, which can then be fed into any standard structure from motion pipeline. This filtering method can be efficiently implemented and scaled to massive datasets as it only requires sparse matrix multiplication. We demonstrate the efficacy of this method on synthetic and real structure from motion datasets and show that it achieves state-of-the-art accuracy and speed in these tasks.

cs.CV↗

arXiv2021

Robust Group Synchronization via Cycle-Edge Message Passing

We propose a general framework for solving the group synchronization problem, where we focus on the setting of adversarial or uniform corruption and sufficiently small noise. Specifically, we apply a novel message passing procedure that uses cycle consistency information in order to estimate the corruption levels of group ratios and consequently solve the synchronization problem in our setting. We first explain why the group cycle consistency information is essential for effectively solving group synchronization problems. We then establish exact recovery and linear convergence guarantees for the proposed message passing procedure under a deterministic setting with adversarial corruption. These guarantees hold as long as the ratio of corrupted cycles per edge is bounded by a reasonable constant. We also establish the stability of the proposed procedure to sub-Gaussian noise. We further establish exact recovery with high probability under a common uniform corruption model.

stat.ML↗

arXiv2021

Ensemble Riemannian Data Assimilation over the Wasserstein Space

In this paper, we present an ensemble data assimilation paradigm over a Riemannian manifold equipped with the Wasserstein metric. Unlike the Eulerian penalization of error in the Euclidean space, the Wasserstein metric can capture translation and difference between the shapes of square-integrable probability distributions of the background state and observations -- enabling to formally penalize geophysical biases in state-space with non-Gaussian distributions. The new approach is applied to dissipative and chaotic evolutionary dynamics and its potential advantages and limitations are highlighted compared to the classic variational and filtering data assimilation approaches under systematic and random errors.

stat.ME↗

arXiv2020

Phase transition in random tensors with multiple independent spikes

Consider a spiked random tensor obtained as a mixture of two components: noise in the form of a symmetric Gaussian $p$-tensor for $p\geq 3$ and signal in the form of a symmetric low-rank random tensor. The latter is defined as a linear combination of $k$ independent symmetric rank-one random tensors, referred to as spikes, with weights referred to as signal-to-noise ratios (SNRs). The entries of the vectors that determine the spikes are i.i.d. sampled from general probability distributions supported on bounded subsets of $\mathbb{R}$. This work focuses on the problem of detecting the presence of these spikes, and establishes the phase transition of this detection problem for any fixed $k \geq 1$. In particular, it shows that for a set of relatively low SNRs it is impossible to distinguish between the spiked and non-spiked Gaussian tensors. Furthermore, in the interior of the complement of this set, where at least one of the $k$ SNRs is relatively high, these two tensors are distinguishable by the likelihood ratio test. In addition, when the total number of low-rank components, $k$, of the $p$-tensor of size $N$ grows in the order $o(N^{(p-2)/4})$ as $N$ tends to infinity, the problem exhibits an analogous phase transition. This theory for spike detection is also shown to imply that recovery of the spikes by the minimum mean square error exhibits the same phase transition. The main methods used in this work arise from the study of mean field spin glass models, where the phase transition thresholds are identified as the critical inverse temperatures distinguishing the high and low-temperature regimes of the free energies. In particular, our result formulates the first full characterization of the high temperature regime for vector-valued spin glass models with independent coordinates.

math.PR↗

arXiv2020

Solving Jigsaw Puzzles By the Graph Connection Laplacian

We propose a novel mathematical framework to address the problem of automatically solving large jigsaw puzzles. This problem assumes a large image, which is cut into equal square pieces that are arbitrarily rotated and shuffled, and asks to recover the original image given the transformed pieces. The main contribution of this work is a method for recovering the rotations of the pieces when both shuffles and rotations are unknown. A major challenge of this procedure is estimating the graph connection Laplacian without the knowledge of shuffles. A careful combination of our proposed method for estimating rotations with any existing method for estimating shuffles results in a practical solution for the jigsaw puzzle problem. Our theory guarantees, in a clean setting, that our basic idea of recovering rotations is robust to some corruption of the connection graph. Numerical experiments demonstrate the competitive accuracy of this solution, its robustness to corruption and, its computational advantage for large puzzles.

cs.CV↗