CUR Decompositions, Similarity Matrices, and Subspace Clustering

A general framework for solving the subspace clustering problem using the CUR decomposition is presented. The CUR decomposition provides a natural way to construct similarity matrices for data that come from a union of unknown subspaces $\mathscr{U}=\underset{i=1}{\overset{M}\bigcup}S_i$. The similarity matrices thus constructed give the exact clustering in the noise-free case. Additionally, this decomposition gives rise to many distinct similarity matrices from a given set of data, which allow enough flexibility to perform accurate clustering of noisy data. We also show that two known methods for subspace clustering can be derived from the CUR decomposition. An algorithm based on the theoretical construction of similarity matrices is presented, and experiments on synthetic and real data are presented to test the method. Additionally, an adaptation of our CUR based similarity matrices is utilized to provide a heuristic algorithm for subspace clustering; this algorithm yields the best overall performance to date for clustering the Hopkins155 motion segmentation dataset.Comment: Approximately 30 pages. Current version contains improved algorithm and numerical experiments from the previous versio

Doubling (Dual) Hahn Polynomials: Classification and Applications

We classify all pairs of recurrence relations in which two Hahn or dual Hahn polynomials with different parameters appear. Such couples are referred to as (dual) Hahn doubles. The idea and interest comes from an example appearing in a finite oscillator model [Jafarov E.I., Stoilova N.I., Van der Jeugt J., J. Phys. A: Math. Theor. 44 (2011), 265203, 15 pages, arXiv:1101.5310]. Our classification shows there exist three dual Hahn doubles and four Hahn doubles. The same technique is then applied to Racah polynomials, yielding also four doubles. Each dual Hahn (Hahn, Racah) double gives rise to an explicit new set of symmetric orthogonal polynomials related to the Christoffel and Geronimus transformations. For each case, we also have an interesting class of two-diagonal matrices with closed form expressions for the eigenvalues. This extends the class of Sylvester-Kac matrices by remarkable new test matrices. We examine also the algebraic relations underlying the dual Hahn doubles, and discuss their usefulness for the construction of new finite oscillator models

Monads, Strings, and M Theory

The recent developments in string theory suggest that the space-time coordinates should be generalized to non-commuting matrices. Postulating this suggestion as the fundamental geometrical principle, we formulate a candidate for covariant second quantized RNS superstrings as a topological field theory in two dimensions. Our construction is a natural non-Abelian extension of the RNS string. It also naturally leads to a model with manifest 11-dimensional covariance, which we conjecture to be a formulation of M theory. The non-commuting space-time coordinates of the strings are further generalized to non-commuting anti-symmetric tensors. The usual space-time picture and the free superstrings appear only in certain special phases of the model. We derive a simple set of algebraic equations, which determine the moduli space of our model. We test some aspects of our conjectual M theory for the case of compactification on $T^2$.Comment: 36 pages, TeX, harvmac, minor corrections with added referenc

Orthogonal Arrays and Legendre Pairs

Well-designed experiments greatly improve test and evaluation. Efficient experiments reduce the cost and time of running tests while improving the quality of the information obtained. Orthogonal Arrays (OAs) and Hadamard matrices are used as designed experiments to glean as much information as possible about a process with limited resources. However, constructing OAs and Hadamard matrices in general is a very difficult problem. Finding Legendre pairs (LPs) results in the construction of Hadamard matrices. This research studies the classification problem of OAs and the existence problem of LPs. In doing so, it makes two contributions to the discipline. First, it improves upon previous classification results of 2-symbol OAs of even-strength t and t+2 columns. Second, it presents previously unknown impossible values for the dimension of the convex hull of all feasible points to the LP problem improving our understanding of its feasible set

Efficiently decodable non-adaptive group testing

We consider the following "efficiently decodable" non-adaptive group testing problem. There is an unknown string x 2 f0; 1gn [x is an element of set {0,1} superscript n] with at most d ones in it. We are allowed to test any subset S [n] [S subset [n] ]of the indices. The answer to the test tells whether xi = 0 [x subscript i = 0] for all i 2 S [i is an element of S] or not. The objective is to design as few tests as possible (say, t tests) such that x can be identifi ed as fast as possible (say, poly(t)-time). Efficiently decodable non-adaptive group testing has applications in many areas, including data stream algorithms and data forensics. A non-adaptive group testing strategy can be represented by a t x n matrix, which is the stacking of all the characteristic vectors of the tests. It is well-known that if this matrix is d-disjunct, then any test outcome corresponds uniquely to an unknown input string. Furthermore, we know how to construct d-disjunct matrices with t = O(d2 [d superscript 2] log n) efficiently. However, these matrices so far only allow for a "decoding" time of O(nt), which can be exponentially larger than poly(t) for relatively small values of d. This paper presents a randomness efficient construction of d-disjunct matrices with t = O(d2 [d superscript 2] log n) that can be decoded in time poly(d) [function composed of] t log2 t + O(t2) [t log superscript 2 t and O (t superscript 2)]. To the best of our knowledge, this is the first result that achieves an efficient decoding time and matches the best known O(d2 log n) [O (d superscript 2 log n)] bound on the number of tests. We also derandomize the construction, which results in a polynomial time deterministic construction of such matrices when d = O(log n= log log n). A crucial building block in our construction is the notion of (d,l)-list disjunct matrices, which represent the more general "list group testing" problem whose goal is to output less than d + l positions in x, including all the (at most d) positions that have a one in them. List disjunct matrices turn out to be interesting objects in their own right and were also considered independently by [Cheraghchi, FCT 2009]. We present connections between list disjunct matrices, expanders, dispersers and disjunct matrices. List disjunct matrices have applications in constructing (d,l)- sparsity separator structures [Ganguly, ISAAC 2008] and in constructing tolerant testers for Reed-Solomon codes in the data stream model. 1 IntroductionDavid & Lucile Packard FoundationCenter for Massive Data Algorithmics (MADALGO)National Science Foundation (U.S.) (Grant CCF-0728645)National Science Foundation (U.S.) (Grant CCF-0347565)National Science Foundation (U.S.) (CAREER Award CCF-0844796

Efficient representation of fully many-body localized systems using tensor networks

We propose a tensor network encoding the set of all eigenstates of a fully many-body localized system in one dimension. Our construction, conceptually based on the ansatz introduced in Phys. Rev. B 94, 041116(R) (2016), is built from two layers of unitary matrices which act on blocks of $\ell$ contiguous sites. We argue this yields an exponential reduction in computational time and memory requirement as compared to all previous approaches for finding a representation of the complete eigenspectrum of large many-body localized systems with a given accuracy. Concretely, we optimize the unitaries by minimizing the magnitude of the commutator of the approximate integrals of motion and the Hamiltonian, which can be done in a local fashion. This further reduces the computational complexity of the tensor networks arising in the minimization process compared to previous work. We test the accuracy of our method by comparing the approximate energy spectrum to exact diagonalization results for the random field Heisenberg model on 16 sites. We find that the technique is highly accurate deep in the localized regime and maintains a surprising degree of accuracy in predicting certain local quantities even in the vicinity of the predicted dynamical phase transition. To demonstrate the power of our technique, we study a system of 72 sites and we are able to see clear signatures of the phase transition. Our work opens a new avenue to study properties of the many-body localization transition in large systems.Comment: Version 2, 16 pages, 16 figures. Larger systems and greater efficienc

Automated Operational Modal Analysis of an arch bridge considering the influence of the parametric methods inputs

Abstract This paper presents a strategy to efficiently obtain modal parameters estimates with parametric methods using a set of values for their input parameters. This is illustrated in detail with the application of the SSI-COV method, by the construction of tri-dimensional stabilization diagrams that consider varying modal orders and varying lengths for the correlation matrices, which are then automatically analyzed using clustering techniques. As case study, it is used an arch bridge built in 1940 with an 80 meter long non-reinforced concrete arch. The developed routines are applied to the datasets collected during a quite extensive ambient vibration test. The work also highlights how the natural frequencies and more particularly the modal damping values may depend on the choice of the parametric method input parameters