arXiv research

This paper finds all prime alternating knots with minimal warping degree two.

Study dynamics of alternating minimization for bilinear regression under large system limits.

TSSM splits neural networks for parallel training with minimal accuracy loss.

Efficiently completes low-rank matrices with nearly linear time complexity.

Alternating minimization represents a widely applicable and empirically successful approach for finding low-rank matrices that best fit the given data. For example, for the problem of low-rank matrix completion, this method is believed to be one of the most accurate and efficient, and formed a major component of the wi…

Minimal grid diagrams for 12-crossing prime knots identified.

This is the third paper in a series devoted to enumerating the prime alternating knots and links. This paper establishes a method for enumerating the prime alternating links. It is shown that one may choose any prime alternating link diagram of a given minimal crossing size and by applications of just two operators (T …

We present theoretical guarantees for an alternating minimization algorithm for the dictionary learning/sparse coding problem. The dictionary learning problem is to factorize vector samples $y^{1},y^{2},\ldots, y^{n}$ into an appropriate basis (dictionary) $A^*$ and sparse vectors $x^{1*},\ldots,x^{n*}$. Our algorithm …

Alternating Minimization is a widely used and empirically successful heuristic for matrix completion and related low-rank optimization problems. Theoretical guarantees for Alternating Minimization have been hard to come by and are still poorly understood. This is in part because the heuristic is iterative and non-conve…

Many applications require recovering a ground truth low-rank matrix from noisy observations of the entries, which in practice is typically formulated as a weighted low-rank approximation problem and solved by non-convex optimization heuristics such as alternating minimization. In this paper, we provide provable recover…

We analyze the performance of alternating minimization for loss functions optimized over two variables, where each variable may be restricted to lie in some potentially nonconvex constraint set. This type of setting arises naturally in high-dimensional statistics and signal processing, where the variables often reflect…

Alternative proof of weak solutions to mean curvature flow using minimizing movements.

We show for an alternating knot the minimal boundary slope of an essential spanning surface is given by the signature plus twice the minimum degree of the Jones polynomial and the maximal boundary slope of an essential spanning surface is given by the signature plus twice the maximum degree of the Jones polynomial. For…

Study shows how fast a specific matrix completion method works.

Minimal grid diagrams found for 13-crossing prime knots.

We present computational results about quasi-alternating knots and links and odd homology obtained by looking at link families in the Conway notation. More precisely, we list quasi-alternating links up to 12 crossings and the first examples of quasi-alternating knots and links with at least two different minimal diagra…

We give a geometric proof of the following result of Juhasz. \emph{Let $a_g$ be the leading coefficient of the Alexander polynomial of an alternating knot $K$. If $|a_g|<4$ then $K$ has a unique minimal genus Seifert surface.} In doing so, we are able to generalise the result, replacing `minimal genus' with `incompress…

This is the first in a series of four papers wherein we enumerate all prime alternating knots and links. In this first paper, we introduce four operators on knots and show that, when used according to very simple rules on the prime alternating knots of n crossings, the set of all prime alternating knots of n+1 crossing…

A meta-learning approach improves the performance of alternating minimization for non-convex optimization problems.

This article is devoted to the study of prime alternating +achiral knots. In the case of arborescent knots, we prove in +AAA Visibility Theorem 5.1, that the symmetry is visible on a certain projection (not necessarily minimal) and that it is realised by a homeomorphism of order 4. In the general case (arborescent or n…

We consider the problem of sparse coding, where each sample consists of a sparse linear combination of a set of dictionary atoms, and the task is to learn both the dictionary elements and the mixing coefficients. Alternating minimization is a popular heuristic for sparse coding, where the dictionary and the coefficient…

Phase retrieval problems involve solving linear equations, but with missing sign (or phase, for complex numbers) information. More than four decades after it was first proposed, the seminal error reduction algorithm of (Gerchberg and Saxton 1972) and (Fienup 1982) is still the popular choice for solving many variants o…

We give a complete proof of results announced by Hirasawa and Sakuma describing explicitly the Kakimizu complex of a non-split, prime, special, alternating link.

Improves deep neural networks using soft labels through alternating minimization.

Families of alternating knots (links) and tangles are studied using as building block the conway defined as the twisting of two strands. The regular representation of knots assumes the projection has the minimal number of overpassings, and the minimal number of conways. The continued fraction associated to rational kno…

Optimized concentric helices minimize the ropelength of non-alternating torus knots.

This paper deals with unsupervised clustering with feature selection. The problem is to estimate both labels and a sparse projection matrix of weights. To address this combinatorial non-convex problem maintaining a strict control on the sparsity of the matrix of weights, we propose an alternating minimization of the Fr…

Proves special alternating knots cannot be decomposed as non-trivial band sums.

We prove a Kauffman-Murasugi-Thistlethwaite theorem for alternating links in thickened surfaces. It states that any reduced alternating diagram of a link in a thickened surface has minimal crossing number, and any two reduced alternating diagrams of the same link have the same writhe. This result is proved more general…

It is shown that there exist alternating non-Montesinos knots whose essential spanning surfaces with maximal and minimal boundary slopes are not realised by the checkerboard surfaces coming from a reduced alternating planar diagram.

We solved a conjecture about braid group quotients being alternating groups.

ALMA improves clustering of multilayer networks.

New tensor completion method converges linearly and is highly practical.

The (global) Lipschitz smoothness condition is crucial in establishing the convergence theory for most optimization methods. Unfortunately, most machine learning and signal processing problems are not Lipschitz smooth. This motivates us to generalize the concept of Lipschitz smoothness condition to the relative smoothn…

Unified rigidity theorem for cyclic and alternating surfaces.

Study sharpens unlinking number bounds for special alternating links.

Alternative proof of simplicial volume bound using area-minimizing sets.

Paper develops efficient AltMin algorithm for SRPCP robust matrix recovery.

Minimal grid diagrams for 15,735 knots with 14 crossings and arc index 14.

New method recovers matrices with nonlinear structures using optimization on Grassmann manifold.

Roberts proved that a family of alternating, arborescent, prime knots each have at least $2^{2n-1}$ distinct minimal genus Seifert surfaces, where $n$ is the genus of the knot in question. We give a subfamily of these knots that have exactly this many minimal genus Seifert surfaces.

Study affine models for alternative risk-free rates and derive caplet pricing formulas.

A new method reduces sample complexity for meta-learning.

We consider the problem of solving mixed random linear equations with $k$ components. This is the noiseless setting of mixed linear regression. The goal is to estimate multiple linear models from mixed samples in the case where the labels (which sample corresponds to which model) are not observed. We give a tractable a…

Paper proves existence of minimal surfaces with alternating multiple zeta values.

Given a class of objects, a pattern theorem is a powerful result describing their structure. We show that alternating knots exhibit a pattern theorem, and use this result to prove a long-standing conjecture that alternating knots grow rare. This is currently the best possible analogue of a pair of theorems on alternati…

We present an objective function for learning with unlabeled data that utilizes auxiliary expectation constraints. We optimize this objective function using a procedure that alternates between information and moment projections. Our method provides an alternate interpretation of the posterior regularization framework (…

We give a local analytic characterization that a minimal surface in the 3-sphere $\, \ES^3 \subset \R^4$ defined by an irreducible cubic polynomial is one of the Lawson's minimal tori. This provides an alternative proof of the result by Perdomo (\emph{Characterization of order 3 algebraic immersed minimal surfaces of $…