arXiv research

Sum-of-squares proofs help solve complex estimation problems.

Paper solves a central question about nonnegative polynomials related to isoparametric polynomials.

New algorithms learn graph structures privately, matching best results.

Paper solves outlier robust mean estimation near breakdown point.

Sum-of-Squares flow improves autoregressive and flow-based density estimation.

New framework reduces sum-of-squares proof degree, speeding up clustering and robust moment estimation.

In recent years, optimization theory has been greatly impacted by the advent of sum of squares (SOS) optimization. The reliance of this technique on large-scale semidefinite programs however, has limited the scale of problems to which it can be applied. In this paper, we introduce DSOS and SDSOS optimization as linear …

Polynomial-time algorithm estimates edge density of random graphs with privacy and robustness.

New algorithm estimates transport maps with nearly optimal error.

Paper introduces a new kernel model for PSD-valued functions with theoretical guarantees and applications.

New perspective on Heegaard splittings using square complexes and combinatorial measurements.

New algorithm predicts tensor entries faster than best rank-r tensor.

New bounds on homological eigenvalues relate to Weil-Petersson length.

The paper develops sum-of-squares relaxations for computing $f$-divergences.

We consider the question of existence of ramified covers over P_1 matching certain prescribed ramification conditions. This problem has already been faced in a number of papers, but we discuss alternative approaches for an existence proof, involving elliptic curves and universal ramified covers with signature. We also …

New flag-no-square 4-manifolds discovered with unique triangulations.

We prove the Chern-Weil formula for SU(n+1)-singular connections over the complement of an embedded oriented surface in smooth four manifolds. The expression of the representation of a number as a sum of nonvanishing squares is given in terms of the representations of a number as a sum of squares. Using the number theo…

In order to model entanglements of polymers in a confined region, we consider the linking numbers and writhes of cycles in random linear embeddings of complete graphs in a cube. Our main results are that for a random linear embedding of $K_n$ in a cube, the mean sum of squared linking numbers and the mean sum of square…

We consider two problems that arise in machine learning applications: the problem of recovering a planted sparse vector in a random linear subspace and the problem of decomposing a random low-rank overcomplete 3-tensor. For both problems, the best known guarantees are based on the sum-of-squares method. We develop new …

Efficiently estimates prediction error in regression with Gaussian covariates under privacy constraints.

We obtain the first polynomial-time algorithm for exact tensor completion that improves over the bound implied by reduction to matrix completion. The algorithm recovers an unknown 3-tensor with $r$ incoherent, orthogonal components in $\mathbb R^n$ from $r\cdot \tilde O(n^{1.5})$ randomly observed entries of the tensor…

In this paper, we generalize the Cao-Yau's gradient estimate for the sum of squares of vector fields up to higher step under assumption of the generalized curvature-dimension inequality. With its applications, by deriving a curvature-dimension inequality, we are able to obtain the Li-Yau gradient estimate for the CR he…

We study a statistical model for the tensor principal component analysis problem introduced by Montanari and Richard: Given a order-$3$ tensor $T$ of the form $T = τ\cdot v_0^{\otimes 3} + A$, where $τ\geq 0$ is a signal-to-noise ratio, $v_0$ is a unit vector, and $A$ is a random noise tensor, the goal is to recover th…

Improves accuracy of SMCI estimators without expanding sum regions.

Explicitly constructed 3XOR instances hard for Sum-of-Squares hierarchy.

Counterexample shows Ito integrand needn't be locally square integrable.

New algorithms recover signals robustly against outliers and heavy-tailed noise.

In the noisy tensor completion problem we observe $m$ entries (whose location is chosen uniformly at random) from an unknown $n_1 \times n_2 \times n_3$ tensor $T$. We assume that $T$ is entry-wise close to being rank $r$. Our goal is to fill in its missing entries using as few observations as possible. Let $n = \max(n…

This letter presents an improved version of diffusion least mean ppower (LMP) algorithm for distributed estimation. Instead of sum of mean square errors, a weighted sum of mean square error is defined as the cost function for global and local cost functions of a network of sensors. The weight coefficients are updated b…

New proof of four squares theorem using projective geometry.

This work improves graph inference using the degree-4 sum-of-squares hierarchy.

Two spectral clustering methods for multi-layer networks are analyzed and compared.

Nearly all Gaussian points in high dimensions lie on a common ellipsoid.

The popular cubic smoothing spline estimate of a regression function arises as the minimizer of the penalized sum of squares $\sum_j(Y_j - μ(t_j))^2 + λ\int_a^b [μ"(t)]^2 dt$, where the data are $t_j,Y_j$, $j=1,..., n$. The minimization is taken over an infinite-dimensional function space, the space of all functions wi…

KSOS improves kernel learning for dynamical systems via global optimization.

Defines a new algebra for singular foliations, extending Schwartz kernels.

This note optimizes distributions using kernel mean embeddings with a new parameterization.

In this paper we prove local analytic hypoellipticity for a degenerate sum of squares of complex vector fields generalizing those of Kohn in "Hypoellipticity and Loss of Derivatives". Kohn's article is to appear in the Annals of Mathematics with an appendix by Derridj and Tartakoff proving local analyticity in that cas…

Algorithm finds a subspace minimizing distances to inliers with outliers.

SOS programming verifies MTW tensor non-negativity for optimal transport maps.

Three algorithms improve starting solutions for clustering problems.

Cross validation residuals extended to GLS models.

We study the map degrees between quasitoric 4-manifolds. Our results rely on Theorems proved by Duan and Wang. We determine the set D (M, N) of all possible map degrees from M to N when M and N are certain quasitoric 4-manifolds. The obtained sets of integers are interesting, e. g. those representable as the sum of two…

New tools in nonlinear random matrices improve understanding of the Sum of Squares hierarchy.

Polynomial-time algorithm estimates mean with bounded covariance using differential privacy.

New method clusters non-spherical Gaussian mixtures with fewer samples and time.

Faster mean estimation with sub-Gaussian error bounds.

GradaGrad adapts learning rate non-monotonically, overcoming AdaGrad's step size decrease.