DOCSLIB.ORG

Tensor Balancing on Statistical Manifold

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Tensor Balancing on Statistical Manifold Tensor Balancing on Statistical Manifold Mahito Sugiyama 1 2 Hiroyuki Nakahara 3 Koji Tsuda 4 5 6 Abstract Given tensor A Statistical manifold (dually at Riemannian manifold) We solve tensor balancing, rescaling an Nth or- Probability der nonnegative tensor by multiplying N ten- distribution P sors of order N − 1 so that every fiber sums to one. This generalizes a fundamental process of Tensor balancing Projection matrix balancing used to compare matrices in a Multistochastic tensor A’ Submanifold (β) wide range of applications from biology to eco- nomics. We present an efficient balancing al- gorithm with quadratic convergence using New- Every ber Projected sums to 1 distribution Pβ ton’s method and show in numerical experiments that the proposed algorithm is several orders of magnitude faster than existing ones. To theo- Figure 1. Overview of our approach. retically prove the correctness of the algorithm, we model tensors as probability distributions in a statistical manifold and realize tensor balanc- Knight, 2016), Hi-C data analysis (Rao et al., 2014; Wu & ing as projection onto a submanifold. The key to Michor, 2016), the Sudoku puzzle (Moon et al., 2009), and our algorithm is that the gradient of the manifold, the optimal transportation problem (Cuturi, 2013; Frogner used as a Jacobian matrix in Newton’s method, et al., 2015; Solomon et al., 2015). An excellent review of can be analytically obtained using the Mobius¨ in- this theory and its applications is given by Idel (2016). version formula, the essential of combinatorial The standard matrix balancing algorithm is the Sinkhorn- mathematics. Our model is not limited to ten- Knopp algorithm (Sinkhorn, 1964; Sinkhorn & Knopp, sor balancing, but has a wide applicability as it 1967; Marshall & Olkin, 1968; Knight, 2008), a special includes various statistical and machine learning case of Bregman’s balancing method (Lamond & Stewart, models such as weighted DAGs and Boltzmann 1981) that iterates rescaling of each row and column until machines. convergence. The algorithm is widely used in the above applications due to its simple implementation and theo- 1. Introduction retically guaranteed convergence. However, the algorithm converges linearly (Soules, 1991), which is prohibitively Matrix balancing is the problem of rescaling a given square slow for recently emerging large and sparse matrices. Al- 2 Rn×n nonnegative matrix A ≥0 to a doubly stochastic ma- though Livne & Golub (2004) and Knight & Ruiz (2013) trix RAS, where every row and column sums to one, by tried to achieve faster convergence by approximating each multiplying two diagonal matrices R and S. This is a step of Newton’s method, the exact Newton’s method with fundamental process for analyzing and comparing matri- quadratic convergence has not been intensively studied yet. ces in a wide range of applications, including input-output Another open problem is tensor balancing, which is a gen- analysis in economics, called the RAS approach (Parikh, eralization of balancing from matrices to higher-order mul- 1979; Miller & Blair, 2009; Lahr & de Mesnard, 2004), tidimentional arrays, or tensors. The task is to rescale an seat assignments in elections (Balinski, 2008; Akartunalı & Nth order nonnegative tensor to a multistochastic tensor, 1National Institute of Informatics 2JST PRESTO 3RIKEN in which every fiber sums to one, by multiplying (N − 1)th Brain Science Institute 4Graduate School of Frontier Sciences, order N tensors. There are some results about mathemat- The University of Tokyo 5RIKEN AIP 6NIMS. Correspondence ical properties of multistochastic tensors (Cui et al., 2014; to: Mahito Sugiyama <[email protected]>. Chang et al., 2016; Ahmed et al., 2003). However, there Proceedings of the 34 th International Conference on Machine is no result for tensor balancing algorithms with guaran- Learning, Sydney, Australia, PMLR 70, 2017. Copyright 2017 teed convergence that transforms a given tensor to a multi- by the author(s). stochastic tensor until now. Tensor Balancing on Statistical Manifold Here we show that Newton’s method with quadratic con- Matrix Constraints for balancing vergence can be applied to tensor balancing while avoid- a a a a η η η η ing solving a linear system on the full tensor. Our strat- a a a a η θ θ θ egy is to realize matrix and tensor balancing as projec- tion onto a dually flat Riemmanian submanifold (Figure 1), a a a a η θ θ θ which is a statistical manifold and known to be the es- a a a a η θ θ θ Invariant sential structure for probability distributions in information geometry (Amari, 2016). Using a partially ordered out- Figure 2. Matrix balancing with two parameters θ and η. come space, we generalize the log-linear model (Agresti, 2012) used to model the higher-order combinations of bi- 2 nary variables (Amari, 2001; Ganmor et al., 2011; Naka- for eachP i; j [n], whereP we normalized entries as pij = hara & Amari, 2002; Nakahara et al., 2003), which allows aij= ij aij so that ij pij = 1. We assume for simplic- us to model tensors as probability distributions in the sta- ity that each entry is strictly larger than zero. The assump- tistical manifold. The remarkable property of our model is tion will be removed in Section 5. that the gradient of the manifold can be analytically com- (t) The key to our approach is that we update θij with i = 1 puted using the Mobius¨ inversion formula (Rota, 1964), the or j = 1 by Newton’s method at each iteration t = 1; 2;::: heart of combinatorial mathematics (Ito, 1993), which en- while fixing θ with i; j =6 1 so that η(t) satisfies the fol- ables us to directly obtain the Jacobian matrix in Newton’s ij ij lowing condition (Figure 2): method. Moreover, we show that (n − 1)N entries for the N (t) − (t) − size n of a tensor are invariant with respect to one of the ηi1 = (n i + 1)=n; η1j = (n j + 1)=n: two coordinate systems of the statistical manifold. Thus Note that the rows and columns sum not to 1 but to 1=n due N−1 the number of equations in Newton’s method is O(n ). to the normalization. The update formula is described as 2 3 2 3 2 3 The remainder of this paper is organized as follows: We θ(t+1) θ(t) η(t) − (n − 1 + 1)=n 6 11 7 6 11 7 6 11 7 begin with a low-level description of our matrix balancing 6 . 7 6 . 7 6 . 7 6 . 7 6 . 7 6 . 7 algorithm in Section 2 and demonstrate its efficiency in nu- 6 7 6 7 6 7 6θ(t+1)7 6θ(t)7 6η(t) − (n − n + 1)=n7 merical experiments in Section 3. To guarantee the correct- 6 1n 7=6 1n 7− J −16 1n 7 ; (3) 6 (t+1)7 6 (t)7 6 (t) − − 7 ness of the algorithm and extend it to tensor balancing, we 6θ21 7 6θ21 7 6η21 (n 2 + 1)=n7 provide theoretical analysis in Section 4. In Section 4.1, we 6 . 7 6 . 7 6 . 7 4 . 5 4 . 5 4 . 5 introduce a generalized log-linear model associated with a (t+1) (t) (t) − − partial order structured outcome space, followed by intro- θn1 θn1 ηn1 (n n + 1)=n ducing the dually flat Riemannian structure in Section 4.2. where J is the Jacobian matrix given as In Section 4.3, we show how to use Newton’s method to (t) @ηij 2 compute projection of a probability distribution onto a sub- J 0 0 = = η f 0g f 0g −n η η 0 0 ; (ij)(i j ) (t) max i;i max j;j ij i j (4) manifold. Finally, we formulate the matrix and tensor bal- @θi0j0 ancing problem in Section 5 and summarize our contribu- which is derived from our theoretical result in Theorem 3. tions in Section 6. Since J is a (2n−1)×(2n−1) matrix, the time complexity of each update is O(n3), which is needed to compute the 2. The Matrix Balancing Algorithm inverse of J. (t+1) (t+1) (t+1) n×n After updating to θ , we can compute p and η Given a nonnegative square matrix A = (a ) 2 R , the ij ij ij ij ≥0 by Equation (2). Since this update does not ensure the task of matrix balancing is to find r; s 2 Rn that satisfy P condition p(t+1) = 1, we again update θ(t+1) as T ij ij P 11 (RAS)1 = 1; (RAS) 1 = 1; (1) (t+1) (t+1) − (t+1) (t+1) θ11 = θ11 log ij pij and recompute pij where R = diag(r) and S = diag(s). The balanced matrix and η(t+1) for each i; j 2 [n]. A0 = RAS is called doubly stochastic, in which each entry ij 0 aij = aijrisj and all the rows and columns sum to one. By iterating the above update process in Equation (3) until The most popular algorithm is the Sinkhorn-Knopp algo- convergence, A = (aij) with aij = npij becomes doubly rithm, which repeats updating r and s as r = 1=(As) and stochastic. s = 1=(AT r). We denote by [n] = f1; 2; : : : ; ng hereafter. In our algorithm, instead of directly updating r and s, we 3. Numerical Experiments update two parameters θ and η defined as X X X X We evaluate the efficiency of our algorithm compared to the log pij = θi0j0 ; ηij = pi0j0 (2) two prominent balancing methods, the standard Sinkhorn- i0≤i j0≤j i0≥i j0≥j Knopp algorithm (Sinkhorn, 1964) and the state-of-the-art Tensor Balancing on Statistical Manifold 10 10 10 10 ations ations 10 r 10 10 r 10 10 10 10 10 – 10 10– 10 Running time (sec.) Running time (sec.) Number of ite Number of ite 10 10– 10 10– 10 50 500 5000 10 50 500 5000 20 100 200 300 20 100 200 300 n n n n Newton (proposed) Sinkhorn BNEWT Newton (proposed) Sinkhorn BNEWT Figure 3.
Load more

Recommended publications
  • The Grassmann Manifold
    The Grassmann Manifold 1. For vector spaces V and W denote by L(V; W ) the vector space of linear maps from V to W . Thus L(Rk; Rn) may be identified with the space Rk£n of k £ n matrices. An injective linear map u : Rk ! V is called a k-frame in V . The set k n GFk;n = fu 2 L(R ; R ): rank(u) = kg of k-frames in Rn is called the Stiefel manifold. Note that the special case k = n is the general linear group: k k GLk = fa 2 L(R ; R ) : det(a) 6= 0g: The set of all k-dimensional (vector) subspaces ¸ ½ Rn is called the Grassmann n manifold of k-planes in R and denoted by GRk;n or sometimes GRk;n(R) or n GRk(R ). Let k ¼ : GFk;n ! GRk;n; ¼(u) = u(R ) denote the map which assigns to each k-frame u the subspace u(Rk) it spans. ¡1 For ¸ 2 GRk;n the fiber (preimage) ¼ (¸) consists of those k-frames which form a basis for the subspace ¸, i.e. for any u 2 ¼¡1(¸) we have ¡1 ¼ (¸) = fu ± a : a 2 GLkg: Hence we can (and will) view GRk;n as the orbit space of the group action GFk;n £ GLk ! GFk;n :(u; a) 7! u ± a: The exercises below will prove the following n£k Theorem 2. The Stiefel manifold GFk;n is an open subset of the set R of all n £ k matrices. There is a unique differentiable structure on the Grassmann manifold GRk;n such that the map ¼ is a submersion.
    [Show full text]
  • On Manifolds of Tensors of Fixed Tt-Rank
    ON MANIFOLDS OF TENSORS OF FIXED TT-RANK SEBASTIAN HOLTZ, THORSTEN ROHWEDDER, AND REINHOLD SCHNEIDER Abstract. Recently, the format of TT tensors [19, 38, 34, 39] has turned out to be a promising new format for the approximation of solutions of high dimensional problems. In this paper, we prove some new results for the TT representation of a tensor U ∈ Rn1×...×nd and for the manifold of tensors of TT-rank r. As a first result, we prove that the TT (or compression) ranks ri of a tensor U are unique and equal to the respective seperation ranks of U if the components of the TT decomposition are required to fulfil a certain maximal rank condition. We then show that d the set T of TT tensors of fixed rank r forms an embedded manifold in Rn , therefore preserving the essential theoretical properties of the Tucker format, but often showing an improved scaling behaviour. Extending a similar approach for matrices [7], we introduce certain gauge conditions to obtain a unique representation of the tangent space TU T of T and deduce a local parametrization of the TT manifold. The parametrisation of TU T is often crucial for an algorithmic treatment of high-dimensional time-dependent PDEs and minimisation problems [33]. We conclude with remarks on those applications and present some numerical examples. 1. Introduction The treatment of high-dimensional problems, typically of problems involving quantities from Rd for larger dimensions d, is still a challenging task for numerical approxima- tion. This is owed to the principal problem that classical approaches for their treatment normally scale exponentially in the dimension d in both needed storage and computa- tional time and thus quickly become computationally infeasable for sensible discretiza- tions of problems of interest.
    [Show full text]
  • Matrices and Tensors
    APPENDIX MATRICES AND TENSORS A.1. INTRODUCTION AND RATIONALE The purpose of this appendix is to present the notation and most of the mathematical tech- niques that are used in the body of the text. The audience is assumed to have been through sev- eral years of college-level mathematics, which included the differential and integral calculus, differential equations, functions of several variables, partial derivatives, and an introduction to linear algebra. Matrices are reviewed briefly, and determinants, vectors, and tensors of order two are described. The application of this linear algebra to material that appears in under- graduate engineering courses on mechanics is illustrated by discussions of concepts like the area and mass moments of inertia, Mohr’s circles, and the vector cross and triple scalar prod- ucts. The notation, as far as possible, will be a matrix notation that is easily entered into exist- ing symbolic computational programs like Maple, Mathematica, Matlab, and Mathcad. The desire to represent the components of three-dimensional fourth-order tensors that appear in anisotropic elasticity as the components of six-dimensional second-order tensors and thus rep- resent these components in matrices of tensor components in six dimensions leads to the non- traditional part of this appendix. This is also one of the nontraditional aspects in the text of the book, but a minor one. This is described in §A.11, along with the rationale for this approach. A.2. DEFINITION OF SQUARE, COLUMN, AND ROW MATRICES An r-by-c matrix, M, is a rectangular array of numbers consisting of r rows and c columns: ¯MM..
    [Show full text]
  • A Some Basic Rules of Tensor Calculus
    A Some Basic Rules of Tensor Calculus The tensor calculus is a powerful tool for the description of the fundamentals in con- tinuum mechanics and the derivation of the governing equations for applied prob- lems. In general, there are two possibilities for the representation of the tensors and the tensorial equations: – the direct (symbolic) notation and – the index (component) notation The direct notation operates with scalars, vectors and tensors as physical objects defined in the three dimensional space. A vector (first rank tensor) a is considered as a directed line segment rather than a triple of numbers (coordinates). A second rank tensor A is any finite sum of ordered vector pairs A = a b + ... +c d. The scalars, vectors and tensors are handled as invariant (independent⊗ from the choice⊗ of the coordinate system) objects. This is the reason for the use of the direct notation in the modern literature of mechanics and rheology, e.g. [29, 32, 49, 123, 131, 199, 246, 313, 334] among others. The index notation deals with components or coordinates of vectors and tensors. For a selected basis, e.g. gi, i = 1, 2, 3 one can write a = aig , A = aibj + ... + cidj g g i i ⊗ j Here the Einstein’s summation convention is used: in one expression the twice re- peated indices are summed up from 1 to 3, e.g. 3 3 k k ik ik a gk ∑ a gk, A bk ∑ A bk ≡ k=1 ≡ k=1 In the above examples k is a so-called dummy index. Within the index notation the basic operations with tensors are defined with respect to their coordinates, e.
    [Show full text]
  • 1.3 Cartesian Tensors a Second-Order Cartesian Tensor Is Defined As A
    1.3 Cartesian tensors A second-order Cartesian tensor is defined as a linear combination of dyadic products as, T Tijee i j . (1.3.1) The coefficients Tij are the components of T . A tensor exists independent of any coordinate system. The tensor will have different components in different coordinate systems. The tensor T has components Tij with respect to basis {ei} and components Tij with respect to basis {e i}, i.e., T T e e T e e . (1.3.2) pq p q ij i j From (1.3.2) and (1.2.4.6), Tpq ep eq TpqQipQjqei e j Tij e i e j . (1.3.3) Tij QipQjqTpq . (1.3.4) Similarly from (1.3.2) and (1.2.4.6) Tij e i e j Tij QipQjqep eq Tpqe p eq , (1.3.5) Tpq QipQjqTij . (1.3.6) Equations (1.3.4) and (1.3.6) are the transformation rules for changing second order tensor components under change of basis. In general Cartesian tensors of higher order can be expressed as T T e e ... e , (1.3.7) ij ...n i j n and the components transform according to Tijk ... QipQjqQkr ...Tpqr... , Tpqr ... QipQjqQkr ...Tijk ... (1.3.8) The tensor product S T of a CT(m) S and a CT(n) T is a CT(m+n) such that S T S T e e e e e e . i1i2 im j1j 2 jn i1 i2 im j1 j2 j n 1.3.1 Contraction T Consider the components i1i2 ip iq in of a CT(n).
    [Show full text]
  • Tensor-Spinor Theory of Gravitation in General Even Space-Time Dimensions
    Physics Letters B 817 (2021) 136288 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Tensor-spinor theory of gravitation in general even space-time dimensions ∗ Hitoshi Nishino a, ,1, Subhash Rajpoot b a Department of Physics, College of Natural Sciences and Mathematics, California State University, 2345 E. San Ramon Avenue, M/S ST90, Fresno, CA 93740, United States of America b Department of Physics & Astronomy, California State University, 1250 Bellflower Boulevard, Long Beach, CA 90840, United States of America a r t i c l e i n f o a b s t r a c t Article history: We present a purely tensor-spinor theory of gravity in arbitrary even D = 2n space-time dimensions. Received 18 March 2021 This is a generalization of the purely vector-spinor theory of gravitation by Bars and MacDowell (BM) in Accepted 9 April 2021 4D to general even dimensions with the signature (2n − 1, 1). In the original BM-theory in D = (3, 1), Available online 21 April 2021 the conventional Einstein equation emerges from a theory based on the vector-spinor field ψμ from a Editor: N. Lambert m lagrangian free of both the fundamental metric gμν and the vierbein eμ . We first improve the original Keywords: BM-formulation by introducing a compensator χ, so that the resulting theory has manifest invariance = =− = Bars-MacDowell theory under the nilpotent local fermionic symmetry: δψ Dμ and δ χ . We next generalize it to D Vector-spinor (2n − 1, 1), following the same principle based on a lagrangian free of fundamental metric or vielbein Tensors-spinors rs − now with the field content (ψμ1···μn−1 , ωμ , χμ1···μn−2 ), where ψμ1···μn−1 (or χμ1···μn−2 ) is a (n 1) (or Metric-less formulation (n − 2)) rank tensor-spinor.
    [Show full text]
  • INTRODUCTION to ALGEBRAIC GEOMETRY 1. Preliminary Of
    INTRODUCTION TO ALGEBRAIC GEOMETRY WEI-PING LI 1. Preliminary of Calculus on Manifolds 1.1. Tangent Vectors. What are tangent vectors we encounter in Calculus? 2 0 (1) Given a parametrised curve α(t) = x(t); y(t) in R , α (t) = x0(t); y0(t) is a tangent vector of the curve. (2) Given a surface given by a parameterisation x(u; v) = x(u; v); y(u; v); z(u; v); @x @x n = × is a normal vector of the surface. Any vector @u @v perpendicular to n is a tangent vector of the surface at the corresponding point. (3) Let v = (a; b; c) be a unit tangent vector of R3 at a point p 2 R3, f(x; y; z) be a differentiable function in an open neighbourhood of p, we can have the directional derivative of f in the direction v: @f @f @f D f = a (p) + b (p) + c (p): (1.1) v @x @y @z In fact, given any tangent vector v = (a; b; c), not necessarily a unit vector, we still can define an operator on the set of functions which are differentiable in open neighbourhood of p as in (1.1) Thus we can take the viewpoint that each tangent vector of R3 at p is an operator on the set of differential functions at p, i.e. @ @ @ v = (a; b; v) ! a + b + c j ; @x @y @z p or simply @ @ @ v = (a; b; c) ! a + b + c (1.2) @x @y @z 3 with the evaluation at p understood.
    [Show full text]
  • DIFFERENTIAL GEOMETRY COURSE NOTES 1.1. Review of Topology. Definition 1.1. a Topological Space Is a Pair (X,T ) Consisting of A
    DIFFERENTIAL GEOMETRY COURSE NOTES KO HONDA 1. REVIEW OF TOPOLOGY AND LINEAR ALGEBRA 1.1. Review of topology. Definition 1.1. A topological space is a pair (X; T ) consisting of a set X and a collection T = fUαg of subsets of X, satisfying the following: (1) ;;X 2 T , (2) if Uα;Uβ 2 T , then Uα \ Uβ 2 T , (3) if Uα 2 T for all α 2 I, then [α2I Uα 2 T . (Here I is an indexing set, and is not necessarily finite.) T is called a topology for X and Uα 2 T is called an open set of X. n Example 1: R = R × R × · · · × R (n times) = f(x1; : : : ; xn) j xi 2 R; i = 1; : : : ; ng, called real n-dimensional space. How to define a topology T on Rn? We would at least like to include open balls of radius r about y 2 Rn: n Br(y) = fx 2 R j jx − yj < rg; where p 2 2 jx − yj = (x1 − y1) + ··· + (xn − yn) : n n Question: Is T0 = fBr(y) j y 2 R ; r 2 (0; 1)g a valid topology for R ? n No, so you must add more open sets to T0 to get a valid topology for R . T = fU j 8y 2 U; 9Br(y) ⊂ Ug: Example 2A: S1 = f(x; y) 2 R2 j x2 + y2 = 1g. A reasonable topology on S1 is the topology induced by the inclusion S1 ⊂ R2. Definition 1.2. Let (X; T ) be a topological space and let f : Y ! X.
    [Show full text]
  • Manifold Reconstruction in Arbitrary Dimensions Using Witness Complexes Jean-Daniel Boissonnat, Leonidas J
    Manifold Reconstruction in Arbitrary Dimensions using Witness Complexes Jean-Daniel Boissonnat, Leonidas J. Guibas, Steve Oudot To cite this version: Jean-Daniel Boissonnat, Leonidas J. Guibas, Steve Oudot. Manifold Reconstruction in Arbitrary Dimensions using Witness Complexes. Discrete and Computational Geometry, Springer Verlag, 2009, pp.37. hal-00488434 HAL Id: hal-00488434 https://hal.archives-ouvertes.fr/hal-00488434 Submitted on 2 Jun 2010 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Manifold Reconstruction in Arbitrary Dimensions using Witness Complexes Jean-Daniel Boissonnat Leonidas J. Guibas Steve Y. Oudot INRIA, G´eom´etrica Team Dept. Computer Science Dept. Computer Science 2004 route des lucioles Stanford University Stanford University 06902 Sophia-Antipolis, France Stanford, CA 94305 Stanford, CA 94305 [email protected] [email protected] [email protected]∗ Abstract It is a well-established fact that the witness complex is closely related to the restricted Delaunay triangulation in low dimensions. Specifically, it has been proved that the witness complex coincides with the restricted Delaunay triangulation on curves, and is still a subset of it on surfaces, under mild sampling conditions. In this paper, we prove that these results do not extend to higher-dimensional manifolds, even under strong sampling conditions such as uniform point density.
    [Show full text]
  • Low-Level Image Processing with the Structure Multivector
    Low-Level Image Processing with the Structure Multivector Michael Felsberg Bericht Nr. 0202 Institut f¨ur Informatik und Praktische Mathematik der Christian-Albrechts-Universitat¨ zu Kiel Olshausenstr. 40 D – 24098 Kiel e-mail: [email protected] 12. Marz¨ 2002 Dieser Bericht enthalt¨ die Dissertation des Verfassers 1. Gutachter Prof. G. Sommer (Kiel) 2. Gutachter Prof. U. Heute (Kiel) 3. Gutachter Prof. J. J. Koenderink (Utrecht) Datum der mundlichen¨ Prufung:¨ 12.2.2002 To Regina ABSTRACT The present thesis deals with two-dimensional signal processing for computer vi- sion. The main topic is the development of a sophisticated generalization of the one-dimensional analytic signal to two dimensions. Motivated by the fundamental property of the latter, the invariance – equivariance constraint, and by its relation to complex analysis and potential theory, a two-dimensional approach is derived. This method is called the monogenic signal and it is based on the Riesz transform instead of the Hilbert transform. By means of this linear approach it is possible to estimate the local orientation and the local phase of signals which are projections of one-dimensional functions to two dimensions. For general two-dimensional signals, however, the monogenic signal has to be further extended, yielding the structure multivector. The latter approach combines the ideas of the structure tensor and the quaternionic analytic signal. A rich feature set can be extracted from the structure multivector, which contains measures for local amplitudes, the local anisotropy, the local orientation, and two local phases. Both, the monogenic signal and the struc- ture multivector are combined with an appropriate scale-space approach, resulting in generalized quadrature filters.
    [Show full text]
  • Hodge Theory
    HODGE THEORY PETER S. PARK Abstract. This exposition of Hodge theory is a slightly retooled version of the author's Harvard minor thesis, advised by Professor Joe Harris. Contents 1. Introduction 1 2. Hodge Theory of Compact Oriented Riemannian Manifolds 2 2.1. Hodge star operator 2 2.2. The main theorem 3 2.3. Sobolev spaces 5 2.4. Elliptic theory 11 2.5. Proof of the main theorem 14 3. Hodge Theory of Compact K¨ahlerManifolds 17 3.1. Differential operators on complex manifolds 17 3.2. Differential operators on K¨ahlermanifolds 20 3.3. Bott{Chern cohomology and the @@-Lemma 25 3.4. Lefschetz decomposition and the Hodge index theorem 26 Acknowledgments 30 References 30 1. Introduction Our objective in this exposition is to state and prove the main theorems of Hodge theory. In Section 2, we first describe a key motivation behind the Hodge theory for compact, closed, oriented Riemannian manifolds: the observation that the differential forms that satisfy certain par- tial differential equations depending on the choice of Riemannian metric (forms in the kernel of the associated Laplacian operator, or harmonic forms) turn out to be precisely the norm-minimizing representatives of the de Rham cohomology classes. This naturally leads to the statement our first main theorem, the Hodge decomposition|for a given compact, closed, oriented Riemannian manifold|of the space of smooth k-forms into the image of the Laplacian and its kernel, the sub- space of harmonic forms. We then develop the analytic machinery|specifically, Sobolev spaces and the theory of elliptic differential operators|that we use to prove the aforementioned decom- position, which immediately yields as a corollary the phenomenon of Poincar´eduality.
    [Show full text]
  • Appendix D: Manifold Maps for SO(N) and SE(N)
    424 Appendix D: Manifold Maps for SO(n) and SE(n) As we saw in chapter 10, recent SLAM implementations that operate with three-dimensional poses often make use of on-manifold linearization of pose increments to avoid the shortcomings of directly optimizing in pose parameterization spaces. This appendix is devoted to providing the reader a detailed account of the mathematical tools required to understand all the expressions involved in on-manifold optimization problems. The presented contents will, hopefully, also serve as a solid base for bootstrap- ping the reader’s own solutions. 1. OPERATOR DEFINITIONS In the following, we will make use of some vector and matrix operators, which are rather uncommon in mobile robotics literature. Since they have not been employed throughout this book until this point, it is in order to define them here. The “vector to skew-symmetric matrix” operator: A skew-symmetric matrix is any square matrix A such that AA= − T . This implies that diagonal elements must be all zeros and off-diagonal entries the negative of their symmetric counterparts. It can be easily seen that any 2× 2 or 3× 3 skew-symmetric matrix only has 1 or 3 degrees of freedom (i.e. only 1 or 3 independent numbers appear in such matrices), respectively, thus it makes sense to parameterize them as a vector. Generating skew-symmetric matrices ⋅ from such vectors is performed by means of the []× operator, defined as: 0 −x 2× 2: = x ≡ []v × () × x 0 x 0 z y (1) − 3× 3: []v = y ≡ z0 − x × z −y x 0 × The origin of the symbol × in this operator follows from its application to converting a cross prod- × uct of two 3D vectors ( x y ) into a matrix-vector multiplication ([]x× y ).
    [Show full text]