TOG 2024: Spectral Total-Variation Processing of Shapes – Theory and Applications

Jonathan Brokman, Martin Burger, Guy Gilboa, ACM Transactions on Graphics, 2024, https://doi.org/10.1145/3641845

We present an analysis of total-variation (TV) on non-Euclidean parameterized surfaces, a natural representation of the shapes used in 3D graphics. Our work explains recent experimental findings in shape spectral TV [Fumero et al., 2020] and adaptive anisotropic spectral TV [Biton and Gilboa, 2022]. A new way to generalize set convexity from the plane to surfaces is derived by characterizing the TV eigenfunctions on surfaces. Relationships between TV, area, eigenvalue, eigenfunctions and their discontinuities are discovered. Further, we expand the shape spectral TV toolkit to include versatile zero-homogeneous flows demonstrated through smoothing and exaggerating filters. Last but not least, we propose the first TV-based method for shape deformation, characterized by deformations along geometrical bottlenecks. We show these bottlenecks to be aligned with eigenfunction discontinuities. This research advances the field of spectral TV on surfaces and its application in 3D graphics, offering new perspectives for shape filtering and deformation.

DXAI: Explaining Classification by Image Decomposition

Elnatan Kadar, Guy Gilboa

arxiv preprint

We propose a new way to explain and to visualize neural network classification through a decomposition-based explainable AI (DXAI). Instead of providing an explanation heatmap, our method yields a decomposition of the image into class-agnostic and class-distinct parts, with respect to the data and chosen classifier. Following a fundamental signal processing paradigm of analysis and synthesis, the original image is the sum of the decomposed parts. We thus obtain a radically different way of explaining classification. The class-agnostic part ideally is composed of all image features which do not posses class information, where the class-distinct part is its complementary. This new visualization can be more helpful and informative in certain scenarios, especially when the attributes are dense, global and additive in nature, for instance, when colors or textures are essential for class distinction.

Critical Points ++: An Agile Point Cloud Importance Measure for Robust Classification, Adversarial Defense and Explainable AI

Yossef Meir Levi, Guy Gilboa

The ability to cope accurately and fast with Out-Of-Distribution (OOD) samples is crucial in real-world safety demanding applications. In this work we first study the interplay between critical points of 3D point clouds and OOD samples. Our findings are that common corruptions and outliers are often interpreted as critical points. We generalize the notion of critical points into importance measures. We show that training a classification network based only on less important points dramatically improves robustness, at a cost of minor performance loss on the clean set. We observe that normalized entropy is highly informative for corruption analysis. An adaptive threshold based on normalized entropy is suggested for selecting the set of uncritical points. Our proposed importance measure is extremely fast to compute. We show it can be used for a variety of applications, such as Explainable AI (XAI), Outlier Removal, Uncertainty Estimation, Robust Classification and Adversarial Defense. We reach SOTA results on the two latter tasks.

arxiv preprint

Generalized Inversion of Nonlinear Operators

Eyal Gofer, Guy Gilboa

Inversion of operators is a fundamental concept in data processing. Inversion of linear operators is well studied, supported by established theory. When an inverse either does not exist or is not unique, generalized inverses are used. Most notable is the Moore-Penrose inverse, widely used in physics, statistics, and various fields of engineering. This work investigates generalized inversion of nonlinear operators.
We first address broadly the desired properties of generalized inverses, guided by the Moore-Penrose axioms. We define the notion for general sets, and then a refinement, termed pseudo-inverse, for normed spaces. We present conditions for existence and uniqueness of a pseudo-inverse and establish theoretical results investigating its properties, such as continuity, its value for operator compositions and projection operators, and others. Analytic expressions are given for the pseudo-inverse of some well-known, non-invertible, nonlinear operators, such as hard- or soft-thresholding and ReLU. We analyze a neural layer and discuss relations to wavelet thresholding.
Next, the Drazin inverse, and a relaxation, are investigated for operators with equal domain and range. We present scenarios where inversion is expressible as a linear combination of forward applications of the operator. Such scenarios arise for classes of nonlinear operators with vanishing polynomials, similar to the minimal or characteristic polynomials for matrices. Inversion using forward applications may facilitate the development of new efficient algorithms for approximating generalized inversion of complex nonlinear operators.

arxiv preprint

EPiC: Ensemble of Partial Point Clouds for Robust Classification, ICCV 2023

Meir Yossef Levi, Guy Gilboa

Accepted to ICCV 2023.

Robust point cloud classification is crucial for real-world applications, as consumer-type 3D sensors often yield partial and noisy data, degraded by various artifacts. In this work we propose a general ensemble framework, based on partial point cloud sampling. Each ensemble member is exposed to only partial input data. Three sampling strategies are used jointly, two local ones, based on patches and curves, and a global one of random sampling. We demonstrate the robustness of our method to various local and global degradations. We show that our framework significantly improves the robustness of top classification netowrks by a large margin. Our experimental setting uses the recently introduced ModelNet-C database by Ren et al.[24], where we reach SOTA both on unaugmented and on augmented data. Our unaugmented mean Corruption Error (mCE) is 0.64 (current SOTA is 0.86) and 0.50 for augmented data (current SOTA is 0.57). We analyze and explain these remarkable results through diversity analysis.

Arxiv preprint

Papers with code

Additive Class Distinction Maps using Branched-GANs

Elnatan Kadar, Jonathan Brokman, Guy Gilboa

Arxiv preprint

We present a new model, training procedure and architecture to create precise maps of distinction between two classes of images. The objective is to comprehend, in pixel-wise resolution, the unique characteristics of a class. These maps can facilitate self-supervised segmentation and objectdetection in addition to new capabilities in explainable AI (XAI). Our proposed architecture is based on image decomposition, where the output is the sum of multiple generative networks (branched-GANs). The distinction between classes is isolated in a dedicated branch. This approach allows clear, precise and interpretable visualization of the unique characteristics of each class. We show how our generic method can be used in several modalities for various tasks, such as MRI brain tumor extraction, isolating cars in aerial photography and obtaining feminine and masculine face features. This is a preliminary report of our initial findings and results.

BASiS: Batch Aligned Spectral Embedding Space, CVPR 2023

Or Streicher, Ido Cohen, Guy Gilboa; Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2023, pp. 10396-10405

CVPR Repository

Arxiv preprint

Graph is a highly generic and diverse representation, suitable for almost any data processing problem. Spectral graph theory has been shown to provide powerful algorithms, backed by solid linear algebra theory. It thus can be extremely instrumental to design deep network building blocks with spectral graph characteristics. For instance, such a network allows the design of optimal graphs for certain tasks or obtaining a canonical orthogonal low-dimensional embedding of the data. Recent attempts to solve this problem were based on minimizing Rayleigh-quotient type losses. We propose a different approach of directly learning the eigensapce. A severe problem of the direct approach, applied in batch-learning, is the inconsistent mapping of features to eigenspace coordinates in different batches. We analyze the degrees of freedom of learning this task using batches and propose a stable alignment mechanism that can work both with batch changes and with graph-metric changes. We show that our learnt spectral embedding is better in terms of NMI, ACC, Grassman distance, orthogonality and classification accuracy, compared to SOTA. In addition, the learning is more stable.

Theoretical Foundations for Pseudo-Inversion of Nonlinear Operators, SSVM-2023

Eyal Gofer, Guy Gilboa, Accepted to SSVM-2023 (oral)

9th International Conference, SSVM 2023, Santa Margherita di Pula, Italy, May 21–25, 2023, Proceedings, Springer LNCS 14009, pp. 29-41, 2023.

arXiv preprint

Springer conference proceedings

Abstract

The Moore-Penrose inverse is widely used in physics, statistics and various fields of engineering. Among other characteristics, it captures well the notion of inversion of linear operators in the case of overcomplete data. In data science, nonlinear operators are extensively used. In this paper we define and characterize the fundamental properties of a pseudo-inverse for nonlinear operators.
The concept is defined broadly. First for general sets, and then a refinement for normed spaces. Our pseudo-inverse for normed spaces yields the Moore-Penrose inverse when the operator is a matrix. We present conditions for existence and uniqueness of a pseudo-inverse and establish theoretical results investigating its properties, such as continuity, its value for operator compositions and projection operators, and others. Analytic expressions are given for the pseudo-inverse of some well-known, non-invertible, nonlinear operators, such as hard- or soft-thresholding and ReLU. Finally, we analyze a neural layer and discuss relations to wavelet thresholding and to regularized loss minimization.

Graph Laplacian for Semi-Supervised Learning, accepted to SSVM-2023

Or Streicher, Guy Gilboa, accepted to SSVM 2023 (oral)

9th International Conference, SSVM 2023, Santa Margherita di Pula, Italy, May 21–25, 2023, Proceedings, Springer LNCS 14009, pp. 250-262, 2023.

Springer conference proceedings

Arxiv preprint

Abatract

Semi-supervised learning is highly useful in common scenarios
where labeled data is scarce but unlabeled data is abundant. The
graph (or nonlocal) Laplacian is a fundamental smoothing operator for
solving various learning tasks. For unsupervised clustering, a spectral embedding
is often used, based on graph-Laplacian eigenvectors. For semisupervised
problems, the common approach is to solve a constrained
optimization problem, regularized by a Dirichlet energy, based on the
graph-Laplacian. However, as supervision decreases, Dirichlet optimization
becomes suboptimal. We therefore would like to obtain a smooth
transition between unsupervised clustering and low-supervised graphbased
classification.
In this paper, we propose a new type of graph-Laplacian which is adapted
for Semi-Supervised Learning (SSL) problems. It is based on both density
and contrastive measures and allows the encoding of the labeled data directly
in the operator. Thus, we can perform successfully semi-supervised
learning using spectral clustering. The benefits of our approach are illustrated
for several SSL problems.

The Underlying Correlated Dynamics in Neural Training, preprint

Rotem Turjeman, Tom Berkov, Ido Cohen, Guy Gilboa

Arxiv preprint

Training of neural networks is a computationally intensive task. The significance of understanding and modeling the training dynamics is growing as increasingly larger networks are being trained. We propose in this work a model based on the correlation of the parameters’ dynamics, which dramatically reduces the dimensionality. We refer to our algorithm as \emph{correlation mode decomposition} (CMD). It splits the parameter space into groups of parameters (modes) which behave in a highly correlated manner through the epochs.
We achieve a remarkable dimensionality reduction with this approach, where networks like ResNet-18, transformers and GANs, containing millions of parameters, can be modeled well using just a few modes. We observe each typical time profile of a mode is spread throughout the network in all layers. Moreover, our model induces regularization which yields better generalization capacity on the test set. This representation enhances the understanding of the underlying training dynamics and can pave the way for designing better acceleration techniques.