Related papers: A Majorization Order on Monomials and Termination …
Let $K$ be a field, and $A=K[a_1,\ldots ,a_n]$ a solvable polynomial algebra in the sense of [K-RW, {\it J. Symbolic Comput.}, 9(1990), 1--26]. It is shown that if $A$ is an $\mathbb{N}$-graded algebra of $({\cal B},d(~))$-type, then $A$…
In this paper, we study the weighted difference substitutions from geometrical views. First, we give the geometric meanings of the weighted difference substitutions, and introduce the concept of convergence of the sequence of substitution…
An algorithm is proposed for solving optimization problems arising in neural network training for supervised learning. The unique feature of the algorithm is the use of an auxiliary loss, in addition to the original loss employed for model…
Sorting algorithms are fundamental to computer science, and their correctness criteria are well understood as rearranging elements of a list according to a specified total order on the underlying set of elements. As mathematical functions,…
We present a rich type system with subtyping for an extension of System F. Our type constructors include sum and product types, universal and existential quantifiers, inductive and coinductive types. The latter two size annotations allowing…
We provide the first generalization error analysis for black-box learning through derivative-free optimization. Under the assumption of a Lipschitz and smooth unknown loss, we consider the Zeroth-order Stochastic Search (ZoSS) algorithm,…
In this paper, we make a contribution to the computation of Gr\"obner bases. For polynomial reduction, instead of choosing the leading monomial of a polynomial as the monomial with respect to which the reduction process is carried out, we…
A recently-established necessary condition for polynomials that preserve the class of entrywise nonnegative matrices of a fixed order is shown to be necessary and sufficient for the class of nonnegative monomial matrices. Along the way, we…
We propose a polynomial time $f$-algorithm (a deterministic algorithm which uses an oracle for factoring univariate polynomials over $\mathbb{F}_q$) for computing an isomorphism (if there is any) of a finite dimensional…
Majorization-minimization schemes are a broad class of iterative methods targeting general optimization problems, including nonconvex, nonsmooth and stochastic. These algorithms minimize successively a sequence of upper bounds of the…
Algorithmic efficiency is essential to reducing energy and time usage for computational problems. Optimizing efficiency is important for tasks involving multiple resources, for example in stochastic calculations where the size of the random…
We give an algorithm for the class of second order unification problems in which second order variables have at most one occurrence.
We formalize a new paradigm for optimality of algorithms, that generalizes worst-case optimality based only on input-size to problem-dependent parameters including implicit ones. We re-visit some existing sorting algorithms from this…
Let $\mathcal{O}$ be an order, that is a commutative ring with $1$ whose additive structure is a free $\mathbb{Z}$-module of finite rank. A generalized number system (GNS for short) over $\mathcal{O}$ is a pair $(p,\mathcal{D} )$ where…
The scalar difference equation $x_{n+1}=f_{n}(x_{n},x_{n-1},...,x_{n-k})$ may exhibit symmetries in its form that allow for reduction of order through substitution or a change of variables. Such form symmetries can be defined generally…
We give a new proof of the simultaneous embedded local uniformization Theorem in zero characteristic for essentially of finite type rings and for quasi excellent rings. The results are a consequence of the simultaneaous monomialization…
We provide a comprehensive view of various phase transitions in random $K$-satisfiability problems solved by stochastic-local-search algorithms. In particular, we focus on the finite-size scaling (FSS) exponent, which is mathematically…
Introduced by M\"uller et al. in their seminal paper \cite{muller}, fractional stochastic dominance (SD) offers a nuanced approach to ordering distributions. In this paper, we propose a fundamentally new framework by replacing the fixed…
The concept of deep dictionary learning has been recently proposed. Unlike shallow dictionary learning which learns single level of dictionary to represent the data, it uses multiple layers of dictionaries. So far, the problem could only be…
We present in this paper a new procedure to saturate a set of clauses with respect to a well-founded ordering on ground atoms such that A < B implies Var(A) {\subseteq} Var(B) for every atoms A and B. This condition is satisfied by any atom…