arXiv research

The paper provides an almost optimal learning and testing algorithm for sparse polynomials.

Volterra and polynomial regression models play a major role in nonlinear system identification and inference tasks. Exciting applications ranging from neuroscience to genome-wide association analysis build on these models with the additional requirement of parsimony. This requirement has high interpretative value, but …

New algorithm recovers sparse measures in polynomial time.

Sparse PCA algorithm improves upon existing methods with better guarantees.

A new method builds sparse polynomial chaos expansions for models with dependent inputs.

Bayesian approach improves sparse PCE for high-dimensional problems.

Detecting correlated trees helps align sparse graphs.

LiPopt uses polynomial optimization to estimate neural network Lipschitz constants efficiently.

Efficiently finds sparse solutions to max-plus equations for convex regression.

New bounds for learning polynomial surrogates with $L_\infty$ guarantees.

This paper improves neural network learning by escaping the NTK regime and efficiently learning sparse polynomials.

The paper shows that g-convex functions on manifolds are sparse.

The challenges for non-intrusive methods for Polynomial Chaos modeling lie in the computational efficiency and accuracy under a limited number of model simulations. These challenges can be addressed by enforcing sparsity in the series representation through retaining only the most important basis terms. In this work, w…

Novel algorithm learns sparse signal representations over topological spaces.

The paper deals with the problem of finding sparse solutions to systems of polynomial equations possibly perturbed by noise. In particular, we show how these solutions can be recovered from group-sparse solutions of a derived system of linear equations. Then, two approaches are considered to find these group-sparse sol…

The paper develops AMP theory for sparse and robust regression with polynomial iterations.

We give a reduction from {\sc clique} to establish that sparse PCA is NP-hard. The reduction has a gap which we use to exclude an FPTAS for sparse PCA (unless P=NP). Under weaker complexity assumptions, we also exclude polynomial constant-factor approximation algorithms.

New algorithm achieves optimal clustering for sparse centers with high dimensions.

This paper optimizes PCE for efficient surrogate modeling in engineering.

Exact causal network discovery is polynomial for sparse networks.

New method uses temperature to control sparse MoE convergence rates.

New algorithms recover sparse tensor principal components efficiently.

New method computes affine normal directions efficiently for sparse polynomials.

The computation of the sparse principal component of a matrix is equivalent to the identification of its principal submatrix with the largest maximum eigenvalue. Finding this optimal submatrix is what renders the problem ${\mathcal{NP}}$-hard. In this work, we prove that, if the matrix is positive semidefinite and its …

New computational lower bounds for clustering and related problems.

New algorithms find half-optimal independent sets in sparse graphs.

We present a novel method for exact hierarchical sparse polynomial regression. Our regressor is that degree $r$ polynomial which depends on at most $k$ inputs, counting at most $\ell$ monomial terms, which minimizes the sum of the squares of its prediction errors. The previous hierarchical sparse specification aligns w…

New method for estimating sparse means in noisy data.

Principal components analysis (PCA) is the optimal linear auto-encoder of data, and it is often used to construct features. Enforcing sparsity on the principal components can promote better generalization, while improving the interpretability of the features. We study the problem of constructing optimal sparse linear a…

Let $f$ be an ordinary polynomial in $\mathbb{C}[z_1,..., z_n]$ with no negative exponents and with no factor of the form $z_1^{α_1}... z_n^{α_n}$ where $α_i$ are non zero natural integer. If we assume in addicting that $f$ is maximally sparse polynomial (that its support is equal to the set of vertices of its Newton p…

The study computes trace fields and minimal polynomials for specific knots and links.

New algorithm reduces sample complexity for sparse linear regression.

New local-search methods close the gap in sparse tensor PCA.

CODE learns ODE dynamics from sparse data, outperforming neural and kernel methods.

New algorithm recovers sparse signals from linearly sparse dictionaries efficiently.

New algorithm recovers sparse signals robustly against Gaussian noise and adaptive adversaries.

New algorithm learns sparse linear MDPs with polynomial interactions, improving sample complexity.

We develop a method to factorize symmetric sparse Boolean matrices efficiently.

New findings support a new community recovery threshold for Stochastic Block Model with many communities.

Dictionary learning is a popular approach for inferring a hidden basis or dictionary in which data has a sparse representation. Data generated from the dictionary A (an n by m matrix, with m > n in the over-complete setting) is given by Y = AX where X is a matrix whose columns have supports chosen from a distribution o…

Conformal prediction improves prediction intervals for PCEs, especially in sparse cases.

Study shows inefficiency of sparse linear regression learning with fewer than Ω(k^2) samples.

It is well known that Sparse PCA (Sparse Principal Component Analysis) is NP-hard to solve exactly on worst-case instances. What is the complexity of solving Sparse PCA approximately? Our contributions include: 1) a simple and efficient algorithm that achieves an $n^{-1/3}$-approximation; 2) NP-hardness of approximatio…

We present a new, far simpler family of counter-examples to Kushnirenko's Conjecture. Along the way, we illustrate a computer-assisted approach to finding sparse polynomial systems with maximally many real roots, thus shedding light on the nature of optimal upper bounds in real fewnomial theory. We use a powerful recen…

New method uses sparse random features for crashworthiness analysis.

We consider the following multi-component sparse PCA problem: given a set of data points, we seek to extract a small number of sparse components with disjoint supports that jointly capture the maximum possible variance. These components can be computed one by one, repeatedly solving the single-component problem and def…

Sparse Bayesian learning improves rational approximations for complex-valued models.

We introduce in this paper the concept of tropical mirror hypersurfaces and we prove a complex tropical localization Theorem which is a version of Kapranov's Theorem \cite{K-00} in tropical geometry. We give a geometric and a topological equivalence between coamoebas of complex algebraic hypersurfaces defined by a maxi…