Related papers: The Legendre Transform in Modern Optimization

Taylor series is a useful mathematical tool when describing and constructing a function. With the series representation, some properties of fractional calculus can be revealed clearly. This paper investigates two typical applications:…

Let $V$ denote a vector space with finite positive dimension. We consider a pair of linear transformations $A : V \to V$ and $A^* : V \to V$ that satisfy (i) and (ii) below: (i) There exists a basis for $V$ with respect to which the matrix…

The classical Cayley transform is a birational map between a quadratic matrix group and its Lie algebra, which was first discovered by Cayley in 1846. Because of its essential role in both pure and applied mathematics, the classical Cayley…

We present a supervised dimensionality reduction technique called Convex Linear Discriminant Analysis (ConvexLDA). The proposed model optimizes a multi-objective cost function by balancing two complementary terms. The first term pulls the…

Transformer encoders contextualize token representations by attending to all other tokens at each layer, leading to quadratic increase in compute effort with the input length. In practice, however, the input text of many NLP tasks can be…

Linear Mode Connectivity (LMC) refers to the phenomenon that performance remains consistent for linearly interpolated models in the parameter space. For independently optimized model pairs from different random initializations, achieving…

In this work, we introduce FLAME, a family of extremely lightweight and capable Time Series Foundation Models, which support both deterministic and probabilistic forecasting via generative probabilistic modeling, thus ensuring both…

Multivariate polynomial optimization is a prevalent model for a number of engineering problems. From a mathematical viewpoint, polynomial optimization is challenging because it is non-convex. The Lasserre's theory, based on semidefinite…

This paper proposes SConvTransform, a Transform dialect extension that provides operations for optimizing 2D convolutions in MLIR. Its main operation, SConvOp, lowers Linalg convolutions into tiled and packed generic operations through a…

Vision Transformers have been tremendously successful in computer vision tasks. However, their large computational, memory, and energy demands are a challenge for edge inference on FPGAs -- a field that has seen a recent surge in demand. We…

This survey of the recent developments in the investigations of a Leavitt path algebra L of an arbitrary graph E over a field K consists of two parts. In the first part describes how very often a single graph property of E implies multiple…

The central topic is this question: is a given $k$-\'etale algebra $\prod_lE_l/k$ the specialization of a given $k$-cover $f:X\rightarrow B$ at some point $t_0\in B(k)$? Our main tool is a {\it twisting lemma} that reduces the problem to…

The great advances of learning-based approaches in image processing and computer vision are largely based on deeply nested networks that compose linear transfer functions with suitable non-linearities. Interestingly, the most frequently…

Let $V$ denote a vector space over C with finite positive dimension. By a {\em Leonard triple} on $V$ we mean an ordered triple of linear operators on $V$ such that for each of these operators there exists a basis of $V$ with respect to…

We present a simple transformation of any linear program or semidefinite program into an equivalent convex optimization problem whose only constraints are linear equations. The objective function is defined on the whole space, making…

It is shown that, under suitable conditions, involving in particular the existence of analytic constants of motion, the presence of Lie point symmetries can ensure the convergence of the transformation taking a vector field (or dynamical…

In the past decade, convolutional neural networks (CNNs) have shown prominence for semantic segmentation. Although CNN models have very impressive performance, the ability to capture global representation is still insufficient, which…

We revisit the results on admissible transformations between normal linear systems of second-order ordinary differential equations with an arbitrary number of dependent variables under several appropriate gauges of the arbitrary elements…

The classical problem of optimal transportation can be formulated as a linear optimization problem on a convex domain: among all joint measures with fixed marginals find the optimal one, where optimality is measured against a cost function.…

We rephrase Monge's optimal transportation (OT) problem with quadratic cost--via a Monge-Amp\`ere equation--as an infinite-dimensional optimization problem, which is in fact a convex problem when the target is a log-concave measure with…