On reconstructing n-point configurations from the distribution of distances or areas

One way to characterize configurations of points up to congruence is by considering the distribution of all mutual distances between points. This paper deals with the question if point configurations are uniquely determined by this distribution. After giving some counterexamples, we prove that this is the case for the vast majority of configurations. In the second part of the paper, the distribution of areas of sub-triangles is used for characterizing point configurations. Again it turns out that most configurations are reconstructible from the distribution of areas, though there are counterexamples.Comment: 21 pages, late

Edge reconstruction of the Ihara zeta function

We show that if a graph $G$ has average degree $\bar d \geq 4$, then the Ihara zeta function of $G$ is edge-reconstructible. We prove some general spectral properties of the edge adjacency operator $T$: it is symmetric for an indefinite form and has a "large" semi-simple part (but it can fail to be semi-simple in general). We prove that this implies that if $\bar d>4$, one can reconstruct the number of non-backtracking (closed or not) walks through a given edge, the Perron-Frobenius eigenvector of $T$ (modulo a natural symmetry), as well as the closed walks that pass through a given edge in both directions at least once. The appendix by Daniel MacDonald established the analogue for multigraphs of some basic results in reconstruction theory of simple graphs that are used in the main text.Comment: 19 pages, 2 pictures, in version 2 some minor changes and now including an appendix by Daniel McDonal

Open Systems Viewed Through Their Conservative Extensions

A typical linear open system is often defined as a component of a larger conservative one. For instance, a dielectric medium, defined by its frequency dependent electric permittivity and magnetic permeability is a part of a conservative system which includes the matter with all its atomic complexity. A finite slab of a lattice array of coupled oscillators modelling a solid is another example. Assuming that such an open system is all one wants to observe, we ask how big a part of the original conservative system (possibly very complex) is relevant to the observations, or, in other words, how big a part of it is coupled to the open system? We study here the structure of the system coupling and its coupled and decoupled components, showing, in particular, that it is only the system's unique minimal extension that is relevant to its dynamics, and this extension often is tiny part of the original conservative system. We also give a scenario explaining why certain degrees of freedom of a solid do not contribute to its specific heat.Comment: 51 page

When Does a Mixture of Products Contain a Product of Mixtures?

We derive relations between theoretical properties of restricted Boltzmann machines (RBMs), popular machine learning models which form the building blocks of deep learning models, and several natural notions from discrete mathematics and convex geometry. We give implications and equivalences relating RBM-representable probability distributions, perfectly reconstructible inputs, Hamming modes, zonotopes and zonosets, point configurations in hyperplane arrangements, linear threshold codes, and multi-covering numbers of hypercubes. As a motivating application, we prove results on the relative representational power of mixtures of product distributions and products of mixtures of pairs of product distributions (RBMs) that formally justify widely held intuitions about distributed representations. In particular, we show that a mixture of products requiring an exponentially larger number of parameters is needed to represent the probability distributions which can be obtained as products of mixtures.Comment: 32 pages, 6 figures, 2 table

The Power of Strong Fourier Sampling: Quantum Algorithms for Affine Groups and Hidden Shifts

Many quantum algorithms, including Shor's celebrated factoring and discrete log algorithms, proceed by reduction to a hidden subgroup problem, in which an unknown subgroup $H$ of a group $G$ must be determined from a quantum state $\psi$ over $G$ that is uniformly supported on a left coset of $H$. These hidden subgroup problems are typically solved by Fourier sampling: the quantum Fourier transform of $\psi$ is computed and measured. When the underlying group is nonabelian, two important variants of the Fourier sampling paradigm have been identified: the weak standard method, where only representation names are measured, and the strong standard method, where full measurement (i.e., the row and column of the representation, in a suitably chosen basis, as well as its name) occurs. It has remained open whether the strong standard method is indeed stronger, that is, whether there are hidden subgroups that can be reconstructed via the strong method but not by the weak, or any other known, method. In this article, we settle this question in the affirmative. We show that hidden subgroups $H$ of the $q$-hedral groups, i.e., semidirect products ${\mathbb Z}_q \ltimes {\mathbb Z}_p$, where $q \mid (p-1)$, and in particular the affine groups $A_p$, can be information-theoretically reconstructed using the strong standard method. Moreover, if $|H| = p/ {\rm polylog}(p)$, these subgroups can be fully reconstructed with a polynomial amount of quantum and classical computation. We compare our algorithms to two weaker methods that have been discussed in the literature—the “forgetful” abelian method, and measurement in a random basis—and show that both of these are weaker than the strong standard method. Thus, at least for some families of groups, it is crucial to use the full power of representation theory and nonabelian Fourier analysis, namely, to measure the high-dimensional representations in an adapted basis that respects the group's subgroup structure. We apply our algorithm for the hidden subgroup problem to new families of cryptographically motivated hidden shift problems, generalizing the work of van Dam, Hallgren, and Ip on shifts of multiplicative characters. Finally, we close by proving a simple closure property for the class of groups over which the hidden subgroup problem can be solved efficiently

Applications of M.G. Krein's Theory of Regular Symmetric Operators to Sampling Theory

The classical Kramer sampling theorem establishes general conditions that allow the reconstruction of functions by mean of orthogonal sampling formulae. One major task in sampling theory is to find concrete, non trivial realizations of this theorem. In this paper we provide a new approach to this subject on the basis of the M. G. Krein's theory of representation of simple regular symmetric operators having deficiency indices (1,1). We show that the resulting sampling formulae have the form of Lagrange interpolation series. We also characterize the space of functions reconstructible by our sampling formulae. Our construction allows a rigorous treatment of certain ideas proposed recently in quantum gravity.Comment: 15 pages; v2: minor changes in abstract, addition of PACS numbers, changes in some keywords, some few changes in the introduction, correction of the proof of the last theorem, and addition of some comments at the end of the fourth sectio

T-uniqueness of some families of k-chordal matroids

We define k-chordal matroids as a generalization of chordal matroids, and develop tools for proving that some k-chordal matroids are T-unique, that is, determined up to isomorphism by their Tutte polynomials. We apply this theory to wheels, whirls, free spikes, binary spikes, and certain generalizations.Postprint (published version