On a Hierarchy of Spectral Isomorphism Invariants

The k-dimensional Weisfeiler-Leman algorithm (\(k\text {-}\mathrm {WL}\)) is a very useful combinatorial tool in graph isomorphism testing. We address the applicability of \(k\text {-}\mathrm {WL}\) to recognition of graph properties. Let G be an input graph with n vertices. We show that, if n is prime, then vertex-transitivity of G can be seen in a straightforward way from the output of \(2\text {-}\mathrm {WL}\) on G and on the vertex-individualized copies of G. This is perhaps the first non-trivial example of using the Weisfeiler-Leman algorithm for recognition of a natural graph property rather than for isomorphism testing. On the other hand, we show that, if n is divisible by 16, then \(k\text {-}\mathrm {WL}\) is unable to distinguish between vertex-transitive and non-vertex-transitive graphs with n vertices unless \(k=\varOmega (\sqrt{n})\).

On the power of combinatorial and spectral invariants

The Weisfeiler-Leman (WL) algorithm is a well-known combinatorial procedure for detecting symmetries in graphs and it is widely used in graph-isomorphism tests. It proceeds by iteratively refining a colouring of vertex tuples. The number of iterations needed to obtain the final output is crucial for the parallelisability of the algorithm. We show that there is a constant k such that every planar graph can be identified (that is, distinguished from every non-isomorphic graph) by the k-dimensional WL algorithm within a logarithmic number of iterations. This generalises a result due to Verbitsky (STACS 2007), who proved the same for 3-connected planar graphs. The number of iterations needed by the k-dimensional WL algorithm to identify a graph corresponds to the quantifier depth of a sentence that defines the graph in the (k+1)-variable fragment C^{k+1} of first-order logic with counting quantifiers. Thus, our result implies that every planar graph is definable with a C^{k+1}-sentence of logarithmic quantifier depth.

In comparison to graphs, combinatorial methods for the isomorphism problem of finite groups are less developed than algebraic ones. To be able to investigate the descriptive complexity of finite groups and the group isomorphism problem, we define the Weisfeiler-Leman algorithm for groups. In fact we define three versions of the algorithm. In contrast to graphs, where the three analogous versions readily agree, for groups the situation is more intricate. For groups, we show that their expressive power is linearly related. We also give descriptions in terms of counting logics and bijective pebble games for each of the versions. In order to construct examples of groups, we devise an isomorphism and non-isomorphism preserving transformation from graphs to groups. Using graphs of high Weisfeiler-Leman dimension, we construct similar but non-isomorphic groups with equal ™(log n)-subgroup-profiles, which nevertheless have Weisfeiler-Leman dimension 3. These groups are nilpotent groups of class 2 and exponent p, they agree in many combinatorial properties such as the combinatorics of their conjugacy classes and have highly similar commuting graphs. The results indicate that the Weisfeiler-Leman algorithm can be more effective in distinguishing groups than in distinguishing graphs based on similar combinatorial constructions.

The Weisfeiler-Leman algorithm and recognition of graph properties

The classical Weisfeiler-Lehman method WL[2] uses edge colors to produce a powerful graph invariant. It is at least as powerful in its ability to distinguish non-isomorphic graphs as the most prominent algebraic graph invariants. It determines not only the spectrum of a graph, and the angles between standard basis vectors and the eigenspaces, but even the angles between projections of standard basis vectors into the eigenspaces. Here, we investigate the combinatorial power of WL[2]. For sufficiently large k, WL[k] determines all combinatorial properties of a graph. Many traditionally used combinatorial invariants are determined by WL[k] for small k. We focus on two fundamental invariants, the number of cycles \(C_p\) of length p, and the number of cliques \(K_p\) of size p. We show that WL[2] determines the number of cycles of lengths up to 6, but not those of length 8. Also, WL[2] does not determine the number of 4-cliques.

On the structure of subsets of the discrete cube with small edge boundary, Discrete Analysis 2018:9, 29 pp. An isoperimetric inequality is a statement that tells us how small the boundary of a set can be given the of the set, for suitable notions of size and boundary. For example, one formulation of the classical isoperimetric inequality in $\mathbb R^n$ is as follows. Given a subset $X$ of $\mathbb R^n$, define the $\epsilon$-_expansion_ of $X$ to be the open set $X_\epsilon=\{y\in\mathbb R^n: d(y,X)<\epsilon\}$. If in addition $X$ is measurable, define the of its boundary to be $\lim\inf_{\epsilon\to 0} \epsilon^{-1}\mu(X_\epsilon\setminus X)$. (If $X$ is a set with a suitably smooth topological boundary $\partial X$, then this turns out to equal the surface measure of $\partial X$.) Then amongst all sets $X$ of a given measure, the one with the smallest boundary is an $n$-dimensional ball. Isoperimetric inequalities have been the focus of a great deal of research, partly for their intrinisic interest, but also because they have numerous applications. One particularly useful one is the _edge-isoperimetric inequality in the discrete cube_. This concerns subsets $X$ of the $n$-dimensional cube $\{0,1\}^n$, which we turn into a graph by joining two points $x$ and $y$ if they differ in exactly one coordinate. The of a set $X$ is simply its cardinality, the _edge-boundary_ of $X$ is defined to be the set of edges between $X$ and its complement, and the of the edge-boundary is the number of such edges. If $|X|=2^d$, then it is known that the edge-boundary is minimized when $X$ is a $d$-dimensional subspace of $\mathbb F_2^n$ generated by $d$ standard basis vectors. More generally, if $|X|=m$, then the edge-boundary is minimized when $X$ is an initial segment in the lexicographical ordering, which is the ordering where we set $x<y$ if $x_i<y_i$ for the first coordinate $i$ where $x_i$ and $y_i$ differ. (This coincides with the ordering we obtain if we think of the sequences as binary representations of integers.) For the two isoperimetric inequalities just mentioned, as well as many others, it is known that the extremal examples provided are essentially the only ones: a subset of $\mathbb R^n$ with a boundary that is as small as possible has to be an $n$-dimensional ball, and a subset of the discrete cube with edge-boundary that is as small as possible has to be an initial segment of the lexicographical ordering, up to the symmetries of the graph. Furthermore, there are _stablity_ results: a set with a boundary that is _almost_ as small as possible must be close to an extremal example. Such a result tells us that the isoperimetric inequalities are robust, in the sense that if you slightly perturb the condition on the set, then you only slightly perturb what the set has to look like. This paper is about an extremely precise stability result for the edge isoperimetric inequality in the discrete cube. There have been a number of papers on such results (see the introduction to the paper for details), but they have been mainly for sets of $2^d$ for some $d$, where the goal is to prove that they must be close to $d$-dimensional subcubes -- that is, subspaces (or their translates) generated by $d$ standard basis vectors. This paper considers sets of arbitrary and proves the following result. Suppose that $X$ is a subset of $\{0,1\}^n$ of $m$. Suppose that the of the edge-boundary of $X$ is at most $g_n(m)+l$, where $g_n(m)$ is the of the edge-boundary of the initial segment $I_m$ of $m$ in the lexicographical order. Then there is an automorphism $\phi$ of $\{0,1\}^n$ (meaning a bijection that takes neighbouring points to neighbouring points) such that $|X\Delta\phi(I_m)|\leq Cl$, where $C$ is an absolute constant. They give an example to show that $C$ must be at least 2, and thus that their result is best possible up to the value of the constant $C$. Previous proofs of stability versions of the edge-isoperimetric inequality in the cube have used Fourier analysis. The proof in this paper uses purely combinatorial methods, such as induction on the dimension, and compressions. To get these methods to work, several interesting ideas are needed, including some new results about the influence of variables.

Representations Theory is used extensively in many of the physical sciences as every physical system has a symmetry group G. Various differential equations determine the vibration of a molecule, and the symmetry group of the molecule acts on the space of solutions of these equations. In this paper we use CH4 (methane) molecule, which has four hydrogen atoms at the corners of a regular tetrahedron, and a carbon atom at the center of the tetrahedron. The four hydrogen atoms in CH4 are permuted by the action of the symmetry group and this action fixes the carbon atom. At each of the 5 vertices, we assign three unit vectors, called the standard basis vectors in directions of the three edges which are joined to the vertex. The symmetry group G of the molecules permutes the 15 standard basis vectors, so we may regard Q15 as a GG By expressing Q15 as a direct sum of irreducible GG-modules, the problem of finding the normal modes of vibration is reduced to that of computing the eigenvectors of some small matrices.

Recently, the gradient method with perturbation (GP) was proposed for metaheuristic methods of solving continuous global optimization problems. Its updating system based on the steepest descent method is chaotic because of its perturbations along the standard basis vectors, which can strengthen the diversification of search. The sufficient condition for its chaoticity was theoretically shown, which implies that two kinds of influence degrees of the perturbations in the updating system should be greater than some constants. In this paper, we extend the updating system of the GP into a more general one for metaheuristic methods, which does not necessarily require the descent direction of the objective function, and which can have perturbations along appropriate orthogonal basis vectors for each problem. Furthermore, since the condition for the chaoticity shown in the previous work is too restricted for practical use, we theoretically show a weaker sufficient condition for the extended system, which is derived by varying the constant lower bounds for the two kinds of influence degrees.

We propose yet another dictionary of orthonormal bases (which has the same tree structure as the popular wavelet packet or local trigonometric dictionaries) adapted to a given ensemble of signals. These orthogonal waveforms are generated by a set of locally adapted versions of the Karhunen-Loeve (KL) transform. The basis vectors in this dictionary represent local features in the time-frequency plane compared to the standard KL basis vectors. Because of the structure of the bases, the best basis selection algorithm of Coifman-Wickerhauser is readily applicable. Moreover, no a priori choice of conjugated quadrature filters or cosine/sine polarity is necessary; it is completely data driven. The computational cost to build this dictionary is comparable to or potentially less than that of the standard KL transform. As an application, we give an example of clustering geophysical acoustic waveforms.

A new implementation of the CMRH method for solving dense linear systems

With the rapid development of information retrieval (IR) systems, online learning to rank (OLR) approaches, which allow retrieval systems to automatically learn best parameters from user interactions, have attracted great research interests in recent years. In OLR, the algorithms usually need to explore some uncertain retrieval results for updating current parameters meanwhile guaranteeing to produce quality retrieval results by exploiting what have already been learned, and the final retrieval results is an interleaved list from both exploratory and exploitative results. However, existing OLR algorithms perform exploration based on either only one stochastic direction or multiple randomly selected stochastic directions, which always involve large variance and uncertainty into the exploration, and may further harm the retrieval quality. Moreover, little historical exploration knowledge is considered when conducting current exploration. In this paper, we propose two OLR algorithms that improve the reliability of the exploration by constructing robust exploratory directions. First, we describe a Dual-Point Dueling Bandit Gradient Descent (DP-DBGD) approach with a Contextual Interleaving (CI) method. In particular, the exploration of DP-DBGD is carefully conducted via two opposite stochastic directions and the proposed CI method constructs a qualified interleaved retrieval result list by taking historical explorations into account. Second, we introduce a Multi-Point Deterministic Gradient Descent (MP-DGD) method that constructs a set of deterministic standard unit basis vectors for exploration. In MP-DGD, each basis direction will be explored and the parameter updating is performed by walking along the combination of exploratory winners from the basis vectors. We conduct experiments on several datasets and show that both DP-DBGD and MP-DGD improve the online learning to rank performance over 10% compared with baseline methods.

Online Variable Sized Covering

Non-negative Matrix Factorization (NMF) has been shown to be effective in providing low-rank, parts-based approximations to canonical datasets comprised of non-negative matrices. The approach involves the factorization of the non-negative matrix A into the product of two non-negative matrices W and H, where the columns of W serve as a set of dictionary vectors for approximating the matrix A. One drawback to this approach is the lack of an exact solution since the problem is not convex in both W and H simultaneously. Previous authors have shown that an exact solution can be achieved by using datasets with specified properties. In this paper we propose a factorial dataset for the use of NMF on patches of a single image. We show that when the multiplicative update is applied to a single image, we are successful in achieving a set of standard basis vectors for the image. We show that by reordering the patches of a specified dataset, the algorithm is successful in achieving exact approximations of single images while preserving the number of standard basis vectors. We use Mean Squared Error (MSE), Peak Signal-to-Noise Ratio (PSNR) and Mean Structured Similarity Index (MSSIM) as measures of the quality of the low rank approximations for a given rank k.

We prove that every finite subspace generated by the linearly ordered idempotent elements in an incline has a unique standard basis.This leads to every finite subspace of a regular incline whose elements are all linearly ordered has a unique standard basis and thereby we disprove the result of Cao that is "Every subspace of a finite incline whose idempotent elements are linearly ordered has a unique standard basis".As an application we exhibit that under certain conditions each vector in a finitely generated subspace of a vector space has a unique decomposition as a linear combination of the standard basis vectors.