arXiv subjects

Michael Perlmutter

Publications and source records attributed to Michael Perlmutter.

At least 37 records · Page 2Linked to original sources

arXiv2023

Understanding Graph Neural Networks with Generalized Geometric Scattering Transforms

The scattering transform is a multilayered wavelet-based deep learning architecture that acts as a model of convolutional neural networks. Recently, several works have introduced generalizations of the scattering transform for non-Euclidean settings such as graphs. Our work builds upon these constructions by introducing windowed and non-windowed geometric scattering transforms for graphs based upon a very general class of asymmetric wavelets. We show that these asymmetric graph scattering transforms have many of the same theoretical guarantees as their symmetric counterparts. As a result, the proposed construction unifies and extends known theoretical results for many of the existing graph scattering architectures. In doing so, this work helps bridge the gap between geometric scattering and other graph neural networks by introducing a large family of networks with provable stability and invariance guarantees. These results lay the groundwork for future deep learning architectures for graph-structured data that have learned filters and also provably have desirable theoretical properties.

stat.ML↗

arXiv2023

Toward Fast and Provably Accurate Near-field Ptychographic Phase Retrieval

Ptychography is an imaging technique which involves a sample being illuminated by a coherent, localized probe of illumination. When the probe interacts with the sample, the light is diffracted and a diffraction pattern is detected. Then the sample (or probe) is shifted laterally in space to illuminate a new area of the sample whilst ensuring sufficient overlap. Near-field Ptychography (NFP) occurs when the sample is placed at a short defocus distance having a large Fresnel number. In this paper, we prove that certain NFP measurements are robustly invertible (up to an unavoidable global phase ambiguity) by constructing a point spread function and physical mask which leads to a well-conditioned lifted linear system. We then apply a block phase retrieval algorithm using weighted angular synchronization and prove that the proposed approach accurately recovers the measured sample. Finally, we also propose using a Wirtinger Flow for NFP problems and numerically evaluate that alternate approach both against our main proposed approach, as well as with NFP measurements for which our main approach does not apply.

math.NA↗

arXiv2022

Taxonomy of Benchmarks in Graph Representation Learning

Graph Neural Networks (GNNs) extend the success of neural networks to graph-structured data by accounting for their intrinsic geometry. While extensive research has been done on developing GNN models with superior performance according to a collection of graph representation learning benchmarks, it is currently not well understood what aspects of a given model are probed by them. For example, to what extent do they test the ability of a model to leverage graph structure vs. node features? Here, we develop a principled approach to taxonomize benchmarking datasets according to a $\textit{sensitivity profile}$ that is based on how much GNN performance changes due to a collection of graph perturbations. Our data-driven analysis provides a deeper understanding of which benchmarking data characteristics are leveraged by GNNs. Consequently, our taxonomy can aid in selection and development of adequate graph benchmarks, and better informed evaluation of future GNN methods. Finally, our approach and implementation in $\texttt{GTaxoGym}$ package are extendable to multiple graph prediction task types and future datasets.

cs.LG↗

arXiv2022

Can Hybrid Geometric Scattering Networks Help Solve the Maximum Clique Problem?

We propose a geometric scattering-based graph neural network (GNN) for approximating solutions of the NP-hard maximum clique (MC) problem. We construct a loss function with two terms, one which encourages the network to find highly connected nodes and the other which acts as a surrogate for the constraint that the nodes form a clique. We then use this loss to train an efficient GNN architecture that outputs a vector representing the probability for each node to be part of the MC and apply a rule-based decoder to make our final prediction. The incorporation of the scattering transform alleviates the so-called oversmoothing problem that is often encountered in GNNs and would degrade the performance of our proposed setup. Our empirical results demonstrate that our method outperforms representative GNN baselines in terms of solution accuracy and inference speed as well as conventional solvers like Gurobi with limited time budgets. Furthermore, our scattering model is very parameter efficient with only $\sim$ 0.1\% of the number of parameters compared to previous GNN baseline models.

cs.LG↗

arXiv2022

Multi-scale Hybridized Topic Modeling: A Pipeline for Analyzing Unstructured Text Datasets via Topic Modeling

We propose a multi-scale hybridized topic modeling method to find hidden topics from transcribed interviews more accurately and efficiently than traditional topic modeling methods. Our multi-scale hybridized topic modeling method (MSHTM) approaches data at different scales and performs topic modeling in a hierarchical way utilizing first a classical method, Nonnegative Matrix Factorization, and then a transformer-based method, BERTopic. It harnesses the strengths of both NMF and BERTopic. Our method can help researchers and the public better extract and interpret the interview information. Additionally, it provides insights for new indexing systems based on the topic level. We then deploy our method on real-world interview transcripts and find promising results.

stat.CO↗

arXiv2022

Learnable Filters for Geometric Scattering Modules

We propose a new graph neural network (GNN) module, based on relaxations of recently proposed geometric scattering transforms, which consist of a cascade of graph wavelet filters. Our learnable geometric scattering (LEGS) module enables adaptive tuning of the wavelets to encourage band-pass features to emerge in learned representations. The incorporation of our LEGS-module in GNNs enables the learning of longer-range graph relations compared to many popular GNNs, which often rely on encoding graph structure via smoothness or similarity between neighbors. Further, its wavelet priors result in simplified architectures with significantly fewer learned parameters compared to competing GNNs. We demonstrate the predictive performance of LEGS-based networks on graph classification benchmarks, as well as the descriptive quality of their learned features in biochemical graph data exploration tasks. Our results show that LEGS-based networks match or outperforms popular GNNs, as well as the original geometric scattering construction, on many datasets, in particular in biochemical domains, while retaining certain mathematical properties of handcrafted (non-learned) geometric scattering.

cs.LG↗

arXiv2022

Overcoming Oversmoothness in Graph Convolutional Networks via Hybrid Scattering Networks

Geometric deep learning has made great strides towards generalizing the design of structure-aware neural networks from traditional domains to non-Euclidean ones, giving rise to graph neural networks (GNN) that can be applied to graph-structured data arising in, e.g., social networks, biochemistry, and material science. Graph convolutional networks (GCNs) in particular, inspired by their Euclidean counterparts, have been successful in processing graph data by extracting structure-aware features. However, current GNN models are often constrained by various phenomena that limit their expressive power and ability to generalize to more complex graph datasets. Most models essentially rely on low-pass filtering of graph signals via local averaging operations, leading to oversmoothing. Moreover, to avoid severe oversmoothing, most popular GCN-style networks tend to be shallow, with narrow receptive fields, leading to underreaching. Here, we propose a hybrid GNN framework that combines traditional GCN filters with band-pass filters defined via geometric scattering. We further introduce an attention framework that allows the model to locally attend over combined information from different filters at the node level. Our theoretical results establish the complementary benefits of the scattering filters to leverage structural information from the graph, while our experiments show the benefits of our method on various learning tasks.

stat.ML↗

arXiv2021

Towards a Taxonomy of Graph Learning Datasets

Graph neural networks (GNNs) have attracted much attention due to their ability to leverage the intrinsic geometries of the underlying data. Although many different types of GNN models have been developed, with many benchmarking procedures to demonstrate the superiority of one GNN model over the others, there is a lack of systematic understanding of the underlying benchmarking datasets, and what aspects of the model are being tested. Here, we provide a principled approach to taxonomize graph benchmarking datasets by carefully designing a collection of graph perturbations to probe the essential data characteristics that GNN models leverage to perform predictions. Our data-driven taxonomization of graph datasets provides a new understanding of critical dataset characteristics that will enable better model evaluation and the development of more specialized GNN models.

cs.LG↗

arXiv2021

A Hybrid Scattering Transform for Signals with Isolated Singularities

The scattering transform is a wavelet-based model of Convolutional Neural Networks originally introduced by S. Mallat. Mallat's analysis shows that this network has desirable stability and invariance guarantees and therefore helps explain the observation that the filters learned by early layers of a Convolutional Neural Network typically resemble wavelets. Our aim is to understand what sort of filters should be used in the later layers of the network. Towards this end, we propose a two-layer hybrid scattering transform. In our first layer, we convolve the input signal with a wavelet filter transform to promote sparsity, and, in the second layer, we convolve with a Gabor filter to leverage the sparsity created by the first layer. We show that these measurements characterize information about signals with isolated singularities. We also show that the Gabor measurements used in the second layer can be used to synthesize sparse signals such as those produced by the first layer.

eess.SP↗

arXiv2021

Scattering Statistics of Generalized Spatial Poisson Point Processes

We present a machine learning model for the analysis of randomly generated discrete signals, modeled as the points of an inhomogeneous, compound Poisson point process. Like the wavelet scattering transform introduced by Mallat, our construction is naturally invariant to translations and reflections, but it decouples the roles of scale and frequency, replacing wavelets with Gabor-type measurements. We show that, with suitable nonlinearities, our measurements distinguish Poisson point processes from common self-similar processes, and separate different types of Poisson point processes.

math.ST↗

arXiv2021

On audio enhancement via online non-negative matrix factorization

We propose a method for noise reduction, the task of producing a clean audio signal from a recording corrupted by additive noise. Many common approaches to this problem are based upon applying non-negative matrix factorization to spectrogram measurements. These methods use a noiseless recording, which is believed to be similar in structure to the signal of interest, and a pure-noise recording to learn dictionaries for the true signal and the noise. One may then construct an approximation of the true signal by projecting the corrupted recording on to the clean dictionary. In this work, we build upon these methods by proposing the use of \emph{online} non-negative matrix factorization for this problem. This method is more memory efficient than traditional non-negative matrix factorization and also has potential applications to real-time denoising.

eess.AS↗

arXiv2021

Modewise Operators, the Tensor Restricted Isometry Property, and Low-Rank Tensor Recovery

Recovery of sparse vectors and low-rank matrices from a small number of linear measurements is well-known to be possible under various model assumptions on the measurements. The key requirement on the measurement matrices is typically the restricted isometry property, that is, approximate orthonormality when acting on the subspace to be recovered. Among the most widely used random matrix measurement models are (a) independent sub-gaussian models and (b) randomized Fourier-based models, allowing for the efficient computation of the measurements. For the now ubiquitous tensor data, direct application of the known recovery algorithms to the vectorized or matricized tensor is awkward and memory-heavy because of the huge measurement matrices to be constructed and stored. In this paper, we propose modewise measurement schemes based on sub-gaussian and randomized Fourier measurements. These modewise operators act on the pairs or other small subsets of the tensor modes separately. They require significantly less memory than the measurements working on the vectorized tensor, provably satisfy the tensor restricted isometry property and experimentally can recover the tensor data from fewer measurements and do not require impractical storage.

math.NA↗

arXiv2021

MagNet: A Neural Network for Directed Graphs

The prevalence of graph-based data has spurred the rapid development of graph neural networks (GNNs) and related machine learning algorithms. Yet, despite the many datasets naturally modeled as directed graphs, including citation, website, and traffic networks, the vast majority of this research focuses on undirected graphs. In this paper, we propose MagNet, a spectral GNN for directed graphs based on a complex Hermitian matrix known as the magnetic Laplacian. This matrix encodes undirected geometric structure in the magnitude of its entries and directional information in their phase. A "charge" parameter attunes spectral information to variation among directed cycles. We apply our network to a variety of directed graph node classification and link prediction tasks showing that MagNet performs well on all tasks and that its performance exceeds all other methods on a majority of such tasks. The underlying principles of MagNet are such that it can be adapted to other spectral GNN architectures.

cs.LG↗

arXiv2021

Phase Retrieval for $L^2([-π,π])$ via the Provably Accurate and Noise Robust Numerical Inversion of Spectrogram Measurements

In this paper, we focus on the approximation of smooth functions $f: [-π, π] \rightarrow \mathbb{C}$, up to an unresolvable global phase ambiguity, from a finite set of Short Time Fourier Transform (STFT) magnitude (i.e., spectrogram) measurements. Two algorithms are developed for approximately inverting such measurements, each with theoretical error guarantees establishing their correctness. A detailed numerical study also demonstrates that both algorithms work well in practice and have good numerical convergence behavior.

math.NA↗

arXiv2019

Inverting Spectrogram Measurements via Aliased Wigner Distribution Deconvolution and Angular Synchronization

We propose a two-step approach for reconstructing a signal ${\bf x}\in\mathbb{C}^d$ from subsampled short-time Fourier transform magnitude (spectogram) measurements: First, we use an aliased Wigner distribution deconvolution approach to solve for a portion of the rank-one matrix ${\bf \widehat{\bf x}}{\bf \widehat{\bf x}}^{*}.$ Second, we use angular syncrhonization to solve for ${\bf \widehat{\bf x}}$ (and then for ${\bf x}$ by Fourier inversion). Using this method, we produce two new efficient phase retrieval algorithms that perform well numerically in comparison to standard approaches and also prove two theorems, one which guarantees the recovery of discrete, bandlimited signals ${\bf x}\in\mathbb{C}^{d}$ from fewer than $d$ STFT magnitude measurements and another which establishes a new class of deterministic coded diffraction pattern measurements which are guaranteed to allow efficient and noise robust recovery.

math.NA↗

arXiv2019

Geometric Scattering on Manifolds

The Euclidean scattering transform was introduced nearly a decade ago to improve the mathematical understanding of the success of convolutional neural networks (ConvNets) in image data analysis and other tasks. Inspired by recent interest in geometric deep learning, which aims to generalize ConvNets to manifold and graph-structured domains, we generalize the scattering transform to compact manifolds. Similar to the Euclidean scattering transform, our geometric scattering transform is based on a cascade of designed filters and pointwise nonlinearities, which enables rigorous analysis of the feature extraction provided by scattering layers. Our main focus here is on theoretical understanding of this geometric scattering network, while setting aside implementation aspects, although we remark that application of similar transforms to graph data analysis has been studied recently in related work. Our results establish conditions under which geometric scattering provides localized isometry invariant descriptions of manifold signals, which are also stable to families of diffeomorphisms formulated in intrinsic manifolds terms. These results not only generalize the deformation stability and local roto-translation invariance of Euclidean scattering, but also demonstrate the importance of linking the used filter structures (e.g., in geometric deep learning) to the underlying manifold geometry, or the data geometry it represents.

stat.ML↗

arXiv2018

Lower Lipschitz Bounds for Phase Retrieval from Locally Supported Measurements

In this short note, we consider the worst case noise robustness of any phase retrieval algorithm which aims to reconstruct all nonvanishing vectors $\mathbf{x} \in \mathbb{C}^d$ (up to a single global phase multiple) from the magnitudes of an arbitrary collection of local correlation measurements. Examples of such measurements include both spectrogram measurements of $\mathbf{x}$ using locally supported windows and masked Fourier transform intensity measurements of $\mathbf{x}$ using bandlimited masks. As a result, the robustness results considered herein apply to a wide range of both ptychographic and Fourier ptychographic imaging scenarios. In particular, the main results imply that the accurate recovery of high-resolution images of extremely large samples using highly localized probes is likely to require an extremely large number of measurements in order to be robust to worst case measurement noise, independent of the recovery algorithm employed. Furthermore, recent pushes to achieve high-speed and high-resolution ptychographic imaging of integrated circuits for process verification and failure analysis will likely need to carefully balance probe design (e.g., their effective time-frequency support) against the total number of measurements acquired in order for their imaging techniques to be stable to measurement noise, no matter what reconstruction algorithms are applied.

math.NA↗

arXiv2018

Large Deviations from the Hydrodynamic Limit for a System with Nearest Neighbor Interactions

We give a new proof of the large deviation principle from the hydrodynamic limit for the Ginzberg-Landau model studied in Donsker and Varadhan (1989) using techniques from the theory of stochastic control and weak convergence methods. The proof is based on characterizing subsequential hydrodynamic limits of controlled diffusions with nearest neighbor interaction that arise from a variational representation of certain Laplace functionals. The approach taken here does not require superexponential probability estimates, estimation of exponential moments, or an analysis of eigenvalue problems, that are central ingredients in previous proofs. Instead, proof techniques are very similar to those used for the law of large number analysis, namely in the proof of convergence to the hydrodynamic limit (cf. Guo, Papanicolaou and Varadhan (1988)). Specifically, the key step in the proof is establishing suitable bounds on relative entropies and Dirichlet forms associated with certain controlled laws. This general approach has the promise to be applicable to other interacting particle systems as well and to the case of non-equilibrium starting configurations, and to infinite volume systems.

math.PR↗