arXiv research

New algorithm for signal estimation in noisy matrix models.

Improves signal detection in non-Gaussian noise using transformed data.

We study the problem of detecting the presence of a single unknown spike in a rectangular data matrix, in a high-dimensional regime where the spike has fixed strength and the aspect ratio of the matrix converges to a finite limit. This setup includes Johnstone's spiked covariance model. We analyze the likelihood ratio …

PLS-SVD struggles with missing data in multimodal datasets, showing a phase transition in performance.

New framework for higher-order singular-value derivatives of rectangular matrices.

The paper studies phase transitions in random matrices and tensor unfolding for detecting signals.

We elaborate on the recent observation that evolution for twist knots simplifies when described in terms of triangular evolution matrix ${\cal B}$, not just its eigenvalues $Λ$, and provide a universal formula for ${\cal B}$, applicable to arbitrary rectangular representation $R=[r^s]$. This expression is in terms of s…

A central problem of random matrix theory is to understand the eigenvalues of spiked random matrix models, introduced by Johnstone, in which a prominent eigenvector (or "spike") is planted into a random matrix. These distributions form natural statistical models for principal component analysis (PCA) problems throughou…

Study reveals efficient recovery of multi-modal signals via Bayesian methods and sequential learning.

This paper analyzes generalization for linear models with spiked covariance structures.

We address the rectangular matrix completion problem by lifting the unknown matrix to a positive semidefinite matrix in higher dimension, and optimizing a nonconvex objective over the semidefinite factor using a simple gradient descent scheme. With $O( μr^2 κ^2 n \max(μ, \log n))$ random observations of a $n_1 \times n…

Improves detection of low-rank signals from noisy data matrices.

Minimal submanifolds in matrix spaces proven for specific ranks.

We claim that the recently discovered universal-matrix precursor for the $F$ functions, which define the differential expansion of colored polynomials for twist and double braid knots, can be extended from rectangular to non-rectangular representations. This case is far more interesting, because it involves multiplicit…

Estimates rank-one spikes from heavy-tailed noise using self-avoiding walks.

Factorization of the differential expansion coefficients for HOMFLY-PT polynomials of double braids, discovered in arXiv:1606.06015 in the case of rectangular representations $R$, is extended to the first non-rectangular representations $R=[2,1]$ and $R=[3,1]$. This increases chances that such factorization will take p…

Study on Langevin dynamics for recovering planted signals in spiked matrix models.

GD and NAG accelerate matrix factorization and neural networks.

Study on signal-plus-noise decomposition in nonlinear spiked random matrices.

Using a low-dimensional parametrization of signals is a generic and powerful way to enhance performance in signal processing and statistical inference. A very popular and widely explored type of dimensionality reduction is sparsity; another type is generative modelling of signal distributions. Generative models based o…

We study the fundamental limits of detecting the presence of an additive rank-one perturbation, or spike, to a Wigner matrix. When the spike comes from a prior that is i.i.d. across coordinates, we prove that the log-likelihood ratio of the spiked model against the non-spiked one is asymptotically normal below a certai…

Paper studies tensor models using random matrix theory.

Simple AMP algorithm robust to adversarial corruption.

The analysis of nonconvex matrix completion has recently attracted much attention in the community of machine learning thanks to its computational convenience. Existing analysis on this problem, however, usually relies on $\ell_{2,\infty}$ projection or regularization that involves unknown model parameters, although th…

Neuronal circuits formed in the brain are complex with intricate connection patterns. Such complexity is also observed in the retina as a relatively simple neuronal circuit. A retinal ganglion cell receives excitatory inputs from neurons in previous layers as driving forces to fire spikes. Analytical methods are requir…

Optimal spectral method found for inhomogeneous spiked Wigner model.

Study shows generative priors improve rank-one matrix recovery with optimal sample complexity.

Improved LDA method for better classification and dimensionality reduction.

A central problem of random matrix theory is to understand the eigenvalues of spiked random matrix models, in which a prominent eigenvector is planted into a random matrix. These distributions form natural statistical models for principal component analysis (PCA) problems throughout the sciences. Baik, Ben Arous and Pé…

Paper proposes a new method for PCA using generative models.

Develops a new point process model for detecting neural spike sequences.

We consider the weak detection problem in a rank-one spiked Wigner data matrix where the signal-to-noise ratio is small so that reliable detection is impossible. We propose a hypothesis test on the presence of the signal by utilizing the linear spectral statistics of the data matrix. The test is data-driven and does no…

The recently suggested KNTZ trick completed the lasting search for exclusive Racah matrices $\bar S$ and $S$ for all rectangular representations and has a potential to help in the non-rectangular case as well. This was the last lacking insight about the structure of differential expansion of (rectangularly-)colored kno…

Consistent model selection for spiked Wigner model via AIC-type criteria.

The (stochastic) gradient descent and the multiplicative update method are probably the most popular algorithms in machine learning. We introduce and study a new regularization which provides a unification of the additive and multiplicative updates. This regularization is derived from an hyperbolic analogue of the entr…

Paper studies S-rectangular DR-RL models for robust reinforcement learning with near-optimal sample complexity.

Rectangular diagrams help analyze foliations in 3-sphere.

A new QDA classifier for high-dimensional data with spiked covariance.

In this work we analyse quantitatively the interplay between the loss landscape and performance of descent algorithms in a prototypical inference problem, the spiked matrix-tensor model. We study a loss function that is the negative log-likelihood of the model. We analyse the number of local minima at a fixed distance …

Adaptive classifier optimizes high-dimensional data with spiked covariance structure.

SGD recovers multiple signal vectors in noisy tensor PCA.

New algorithms improve Bayesian linear regression with spike-and-slab priors.

Algorithm finds isotropy subgroups of orthogonal similarity on symmetric matrices.

Computes isotropy subgroups of orthogonal matrices acting on Hermitian matrices.

New algorithms sample spike-and-slab priors efficiently in high dimensions.

Study eigenvalues and eigenvectors in neural networks, focusing on signal propagation.

New framework predicts AMP behavior in spiked models for finite iterations.

Improved bounds for knot crossings in different mosaic patterns.