Related papers: Principal gradient schemes have regular reduced cl…
In this paper we give a description of the first order deformation space of a regular embedding of reduced algebraic schemes. We compare our result with results of Ran (in particular [Ran, Prop. 1.3]).
The criterion for an affine primary algebra over the field to be integral, is proven. Using this criterion we give a simple proof that Hilbert scheme of 0-dimensional subschemes of length $l$ of nonsingular $d$-dimensional algebraic variety…
We consider the class of all commutative reduced rings for which there exists a finite subset T of A such that all projections on quotients by prime ideals of A are surjective when restricted to T. A complete structure theorem is given for…
We develop subgradient- and gradient-based methods for minimizing strongly convex functions under a notion which generalizes the standard Euclidean strong convexity. We propose a unifying framework for subgradient methods which yields two…
Gradient descent can be surprisingly good at optimizing deep neural networks without overfitting and without explicit regularization. We find that the discrete steps of gradient descent implicitly regularize models by penalizing gradient…
Classes of polynomial differential equations of degree n are considered. An explicit upper bound on the size of the coefficients are given which implies that each equation in the class has exactly n complex periodic solutions. In most of…
This paper is partly a report on current knowledge concerning the structure of (generic) quantized coordinate rings and their prime spectra, and partly propaganda in support of the conjecture that since these algebras share many common…
We study primary submodules and primary decompositions from a differential and computational point of view. Our main theoretical contribution is a general structure theory and a representation theorem for primary submodules of an arbitrary…
We describe the equations of the Rees algebra R(I) of an equimultiple ideal I of deviation one, provided that I has a reduction J generated by a regular sequence and such that the initial forms of the elements of this sequence, except…
Proper regularization is critical for speeding up training, improving generalization performance, and learning compact models that are cost efficient. We propose and analyze regularized gradient descent algorithms for learning shallow…
We introduce a generalization of the conventional renormalization schemes used in dimensional regularization, which illuminates the renormalization scheme and scale ambiguities of pQCD predictions, exposes the general pattern of…
We review the construction (due to Brenner--Schr\"oer) of the Proj scheme associated with a ring graded by a finitely generated abelian group. This construction generalizes the well-known Grothendieck Proj construction for…
In the famous paper of Deligne and Mumford, they proved that a proper hyperbolic curve over a discrete valuation field has stable reduction if and only if the Jacobian variety of the curve has stable reduction in the case where the residue…
In this paper, two new subspace minimization conjugate gradient methods based on $p - $regularization models are proposed, where a special scaled norm in $p - $regularization model is analyzed. Different choices for special scaled norm lead…
Gradient regularization (GR) is a method that penalizes the gradient norm of the training loss during training. While some studies have reported that GR can improve generalization performance, little attention has been paid to it from the…
Efforts to understand the generalization mystery in deep learning have led to the belief that gradient-based optimization induces a form of implicit regularization, a bias towards models of low "complexity." We study the implicit…
A new DRP scheme is built, which enables us to minimize the error due to the finite difference approximation, by means of an equivalent matrix equation.
We establish a general principle which states that regularizing an inverse problem with a convex function yields solutions which are convex combinations of a small number of atoms. These atoms are identified with the extreme points and…
Jordan Normal Forms serve as excellent representatives of conjugacy classes of matrices over closed fields. Once we knows normal forms, we can compute functions of matrices, their main invariant, etc. The situation is much more complicated…
When the gate set has continuous parameters, synthesizing a unitary operator as a quantum circuit is always possible using exact methods, but finding minimal circuits efficiently remains a challenging problem. The landscape is very…