arXiv research

New algorithms for hypothesis testing in high-dimensional data are shown to be effective under various noisy conditions.

New algorithms detect categorical structures in high-dimensional data.

New spectral clustering method for graphs with uneven node degrees.

We propose a new stochastic coordinate descent method for minimizing the sum of convex functions each of which depends on a small number of coordinates only. Our method (APPROX) is simultaneously Accelerated, Parallel and PROXimal; this is the first time such a method is proposed. In the special case when the number of…

Learn low-degree functions with few random queries.

Survey on using low-degree polynomials to assess statistical tasks complexity.

Paper connects free-energy and low-degree hardness in high-dimensional statistics.

Study mass transport in low-diffusivity using Lagrangian coordinates.

Sublinear algorithms detect cliques in graphs with high probability.

Flat coordinates found for algebraic Frobenius manifolds in low dimensions.

Optimizes CM for stochastic convex optimization with progressive precision.

Paper speeds up IoT device detection and data decoding.

Paper proves robust M-estimators' coordinates' normality in high dimensions.

Low-degree method fails to predict robust subspace recovery problem.

New findings on computational limits for estimating hidden structures.

For each simple Lie algebra $\mathfrak{g}$ (excluding, for trivial reasons, type ${\sf C}$) we find the lowest possible degree of an invariant second-order PDE over the adjoint variety in $\mathbb{P}\mathfrak{g}$, a homogeneous contact manifold. Here a PDE $F(x^i,u,u_i,u_{ij})=0$ has degree $\le d$ if $F$ is a polynomi…

Manifold embedding algorithms map high-dimensional data down to coordinates in a much lower-dimensional space. One of the aims of dimension reduction is to find intrinsic coordinates that describe the data manifold. The coordinates returned by the embedding algorithm are abstract, and finding their physical or domain-r…

Novel deep learning method predicts reaction coordinates and future MD trajectories.

The paper models financial order books using geometric shears and directional liquidity.

The systole function has a universal index gap on moduli spaces.

This paper focuses on coordinate update methods, which are useful for solving problems involving large or high-dimensional datasets. They decompose a problem into simple subproblems, where each updates one, or a small block of, variables while fixing others. These methods can deal with linear and nonlinear mappings, sm…

In Physics and in Mathematics $\mathbb{Z}_2^n$-gradings, $n>1$, appear in various fields. The corresponding sign rule is determined by the `scalar product' of the involved $\mathbb{Z}_2^n$-degrees. The $\mathbb{Z}_2^n$-Supergeometry exhibits challenging differences with the classical one: nonzero degree even coordinate…

Study disproves conjecture about low-degree polynomials in hypothesis testing.

In Physics and in Mathematics $\mathbb{Z}_2^n$-gradings, $n \geq 2$, do appear quite frequently. The corresponding sign rules are determined by the `scalar product' of the involved $\mathbb{Z}_2^n$-degrees. The present paper is the first of a series on $\mathbb{Z}_2^n$-Supergeometry. The new theory exhibits challenging…

Gradient Descent with Projection learns low-degree polynomials efficiently.

Smooth $\mathbb{Z}_2^n$-supermanifolds have been introduced and studied recently. The corresponding sign rule is given by the "scalar product" of the involved $\mathbb{Z}_2^n$-degrees. It exhibits interesting changes in comparison with the sign rule using the parity of the total degree. With the new rule, nonzero degre…

Space exploration missions have seen use of increasingly sophisticated robotic systems with ever more autonomy. Deep learning promises to take this even a step further, and has applications for high-level tasks, like path planning, as well as low-level tasks, like motion control, which are critical components for missi…

The degree-$d$ Chow parameters of a Boolean function $f: \{-1,1\}^n \to \mathbb{R}$ are its degree at most $d$ Fourier coefficients. It is well-known that degree-$d$ Chow parameters uniquely characterize degree-$d$ polynomial threshold functions (PTFs) within the space of all bounded functions. In this paper, we prove …

Two-layer NN with channel attention learns low-degree spherical polynomials efficiently.

New work shows FP potential monotonicity equals low-degree polynomial estimators limits.

Paper develops new patterns for unique matrix completions.

Paper proves first non-trivial PTF testing lower bounds for NGCA.

RC flow learns molecular kinetics in low dimensions.

Study tackles nonlinear factor models with unknown monotone links from incomplete and noisy data.

Hashing has been widely adopted for large-scale data retrieval in many domains, due to its low storage cost and high retrieval speed. Existing cross-modal hashing methods optimistically assume that the correspondence between training samples across modalities are readily available. This assumption is unrealistic in pra…

A new method for high-dimensional Bayesian optimization.

These notes survey and explore an emerging method, which we call the low-degree method, for predicting and understanding statistical-versus-computational tradeoffs in high-dimensional inference problems. In short, the method posits that a certain quantity -- the second moment of the low-degree likelihood ratio -- gives…

We introduce the concept of a graded bundle which is a natural generalization of the concept of a vector bundle and whose standard examples are higher tangent bundles T^nQ playing a fundamental role in higher order Lagrangian formalisms. Graded bundles are graded manifolds in the sense that we can choose an atlas whose…

The paper geometrically characterizes graded manifolds and proves the Frobenius theorem.

New algorithm learns PTFs with noisy data efficiently.

Detects dense subhypergraphs in random hypergraphs using low-degree polynomials.

New method explains computational barriers in high-dimensional statistical models.

Christoffel function characterizes the corruption a bounded-degree certificate cannot remove in robust halfspace learning.

This article describes a multivariate polynomial regression method where the uncertainty of the input parameters are approximated with Gaussian distributions, derived from the central limit theorem for large weighted sums, directly from the training sample. The estimated uncertainties can be propagated into the optimal…

GCNs favor high-degree nodes, leading to biased performance; a new method mitigates this.

Algorithm learns polynomial transformations of Gaussian distributions.

Study uses reinforcement learning to optimize metachronal paddling at low Reynolds number.

We propose a novel method to embed a functional magnetic resonance imaging (fMRI) dataset in a low-dimensional space. The embedding optimally preserves the local functional coupling between fMRI time series and provides a low-dimensional coordinate system for detecting activated voxels. To compute the embedding, we bui…