Product (mathematics)

About: Product (mathematics) is a research topic. Over the lifetime, 44382 publications have been published within this topic receiving 377809 citations.

A PAC-Bayesian Approach to Spectrally-Normalized Margin Bounds for Neural Networks

Abstract: We present a generalization bound for feedforward neural networks in terms of the product of the spectral norm of the layers and the Frobenius norm of the weights. The generalization bound is derived using a PAC-Bayes analysis.

Letter to the Editor—Additive Utilities with Incomplete Product Sets: Application to Priorities and Assignments

Abstract: A previous paper discusses the notion of additive utilities in the expected-utility context when the set of consequences X can be written as the product of n other sets: X = X1 × X2 × ⋯ × Xn. The present paper continues this investigation for cases where X is a subset of the product of n other sets: X ⊂ X1 × X2 × ⋯ × Xn. Under a finiteness restriction, an additivity axiom leads to an additive utility representation for the elements in X. The potential usefulness of the theory in connection with incomplete product sets is demonstrated with priority orderings and assignments.

Limit theory for multivariate sample extremes

Abstract: Let $$\{ (X_n^{(1)} ),...,X_n^{(k)} ,{\text{ }}n\} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \geqslant } 1\} $$ be k-dimensional iid random vectors. Necessary and sufficient conditions are found for the weak convergence of the maxima $$\left\{ {\mathop {\text{V}}\limits_{j{\text{ = 1}}}^n X_j^{(1)} ,...,\mathop {\text{V}}\limits_{j{\text{ = 1}}}^n X_j^{(k)} } \right\}$$ suitably normed to a non-degenerate limit df. The class of such limits is specified and conditions stated for the limit joint df to be a product of marginal df's. Some results are presented concerning extremal processes generated by multivariate df's.

Group Actions, Homeomorphisms, and Matching: A General Framework

Abstract: This paper constructs metrics on the space of images {\cal I} defined as orbits under group actions {\cal G}. The groups studied include the finite dimensional matrix groups and their products, as well as the infinite dimensional diffeomorphisms examined in Trouve (1999, Quaterly of Applied Math.) and Dupuis et al. (1998). Quaterly of Applied Math.). Left-invariant metrics are defined on the product {\cal G} \times {\cal I} thus allowing the generation of transformations of the background geometry as well as the image values. Examples of the application of such metrics are presented for rigid object matching with and without signature variation, curves and volume matching, and structural generation in which image values are changed supporting notions such as tissue creation in carrying one image to another.

Noncommutative symmetric functions

Abstract: This paper presents a noncommutative theory of symmetric functions, based on the notion of quasi-determinant. We begin with a formal theory, corresponding to the case of symmetric functions in an infinite number of independent variables. This allows us to endow the resulting algebra with a Hopf structure, which leads to a new method for computing in descent algebras. It also gives unified reinterpretation of a number of classical constructions. Next, we study the noncommutative analogs of symmetric polynomials. One arrives at different constructions, according to the particular kind of application under consideration. For example, when a polynomial with noncommutative coefficients in one central variable is decomposed as a product of linear factors, the roots of these factors differ from those of the expanded polynomial. Thus, according to whether one is interested in the construction of a polynomial with given roots or in the expansion of a product of linear factors, one has to consider two distinct specializations of the formal symmetric functions. A third type appears when one looks for a noncommutative generalization of applications related to the notion of characteristic polynomial of a matrix. This construction can be applied, for instance, to the noncommutative matrices formed by the generators of the universal enveloping algebra $U(gl_n)$ or of