September 2017, 9(3): 335-390. doi: 10.3934/jgm.2017014

Geometry of matrix decompositions seen through optimal transport and information geometry

Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, SE-412 96 Gothenburg, Sweden

* Corresponding author

Received  January 2016 Revised  August 2016 Published  June 2017

Fund Project: This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 661482, and from the Swedish Foundation for Strategic Research under grant agreement ICA12-0052

The space of probability densities is an infinite-dimensional Riemannian manifold, with Riemannian metrics in two flavors: Wasserstein and Fisher-Rao. The former is pivotal in optimal mass transport (OMT), whereas the latter occurs in information geometry--the differential geometric approach to statistics. The Riemannian structures restrict to the submanifold of multivariate Gaussian distributions, where they induce Riemannian metrics on the space of covariance matrices.

Here we give a systematic description of classical matrix decompositions (or factorizations) in terms of Riemannian geometry and compatible principal bundle structures. Both Wasserstein and Fisher-Rao geometries are discussed. The link to matrices is obtained by considering OMT and information geometry in the category of linear transformations and multivariate Gaussian distributions. This way, OMT is directly related to the polar decomposition of matrices, whereas information geometry is directly related to the $QR$, Cholesky, spectral, and singular value decompositions. We also give a coherent description of gradient flow equations for the various decompositions; most flows are illustrated in numerical examples.

The paper is a combination of previously known and original results. As a survey it covers the Riemannian geometry of OMT and polar decompositions (smooth and linear category), entropy gradient flows, and the Fisher-Rao metric and its geodesics on the statistical manifold of multivariate Gaussian distributions. The original contributions include new gradient flows associated with various matrix decompositions, new geometric interpretations of previously studied isospectral flows, and a new proof of the polar decomposition of matrices based an entropy gradient flow.

Citation: Klas Modin. Geometry of matrix decompositions seen through optimal transport and information geometry. Journal of Geometric Mechanics, 2017, 9 (3) : 335-390. doi: 10.3934/jgm.2017014
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show all references

References:
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M. Adler, On a trace functional for formal pseudo-differential operators and the symplectic structure of the Korteweg-deVries type equations, Invent. Math., 50 (1978), 219-248. doi: 10.1007/BF01410079.

[2]

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S. Amari and H. Nagaoka, Methods of Information Geometry, Amer. Math. Soc., Providence, RI, 2000.

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S. AngenentS. Haker and A. Tannenbaum, Minimizing flows for the Monge-Kantorovich problem, SIAM J. Math. Anal., 35 (2003), 61-97. doi: 10.1137/S0036141002410927.

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F. Barbaresco, Information geometry of covariance matrix: Cartan-siegel homogeneous bounded domains, mostow/berger fibration and frechet median, in Matrix Information Geometry, Springer, 2013,199-255. doi: 10.1007/978-3-642-30232-9_9.

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J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the monge-kantorovich mass transfer problem, Numer. Math., 84 (2000), 375-393. doi: 10.1007/s002110050002.

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J. -D. Benamou, Y. Brenier and A. Oberman, Advances in Numerical Optimal Transportation, Technical Report 15w5067, Banff International Research Station, 2015.

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A. M. BlochR. W. Brockett and T. S. Ratiu, Completely integrable gradient flows, Comm. Math. Phys., 147 (1992), 57-74. doi: 10.1007/BF02099528.

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Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math., 44 (1991), 375-417. doi: 10.1002/cpa.3160440402.

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M. T. Chu, The generalized Toda flow, the QR algorithm and the center manifold theory, SIAM J. Alg. Discrete Meth., 5 (1984), 187-201. doi: 10.1137/0605020.

[24]

M. T. Chu, Matrix differential equations: A continuous realization process for linear algebra problems, Nonlin. Anal.: Theor. Meth. & Appl., 18 (1992), 1125-1146. doi: 10.1016/0362-546X(92)90157-A.

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[26]

M. T. Chu, Scaled Toda-like flows, Linear Algebra Appl., 215 (1995), 261-273. doi: 10.1016/0024-3795(93)00091-D.

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M. T. Chu, Linear algebra algorithms as dynamical systems, Acta Numer., 17 (2008), 1-86. doi: 10.1017/S0962492906340019.

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M. T. Chu and K. R. Driessel, Constructing symmetric nonnegative matrices with prescribed eigenvalues by differential equations, SIAM J. Math. Anal., 22 (1991), 1372-1387. doi: 10.1137/0522088.

[29]

M. T. Chu and L. K. Norris, Isospectral flows and abstract matrix factorizations, SIAM J. Numer. Anal., 25 (1988), 1383-1391. doi: 10.1137/0725080.

[30]

B. Clarke, The metric geometry of the manifold of Riemannian metrics over a closed manifold, Calc. Var. PDE, 39 (2010), 533-545. doi: 10.1007/s00526-010-0323-5.

[31]

B. Clarke, The completion of the manifold of Riemannian metrics, J. Differential Equations, 93 (2013), 203-268. doi: 10.4310/jdg/1361800866.

[32]

P. DeiftJ. DemmelL.-C. Li and C. Tomei, The bidiagonal singular value decomposition and Hamiltonian mechanics, SIAM J. Numer. Anal., 28 (1991), 1463-1516. doi: 10.1137/0728076.

[33]

P. DeiftT. Nanda and C. Tomei, Ordinary differential equations and the symmetric eigenvalue problem, SIAM J. Numer. Anal., 20 (1983), 1-22. doi: 10.1137/0720001.

[34]

P. DeiftL. LiT. Nanda and C. Tomei, The Toda flow on a generic orbit is integrable, Comm. Pure Appl. Math., 39 (1986), 183-232. doi: 10.1002/cpa.3160390203.

[35]

D. G. Ebin, On the space of Riemannian metrics, Bull. Amer. Math. Soc., 74 (1968), 1001-1003. doi: 10.1090/S0002-9904-1968-12115-9.

[36]

D. G. Ebin and J. E. Marsden, Groups of diffeomorphisms and the notion of an incompressible fluid, Ann. of Math., 92 (1970), 102-163. doi: 10.2307/1970699.

[37]

R. A. Fisher, On the mathematical foundations of theoretical statistics, Breakthroughs in Statistics: Part of the series Springer Series in Statistics, (1992), 11-44. doi: 10.1007/978-1-4612-0919-5_2.

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H. Flaschka, The Toda lattice. Ⅱ. existence of integrals, Physical Review B, 9 (1974), 1924-1925. doi: 10.1103/PhysRevB.9.1924.

[39]

D. S. Freed and D. Groisser, The basic geometry of the manifold of Riemannian metrics and of its quotient by the diffeomorphism group, Michigan Math. J., 36 (1989), 323-344. doi: 10.1307/mmj/1029004004.

[40]

T. Friedrich, Die Fisher-information und symplektische strukturen, Math. Nachr., 153 (1991), 273-296. doi: 10.1002/mana.19911530125.

[41]

N. H. Getz and J. E. Marsden, Dynamical methods for polar decomposition and inversion of matrices, Linear Algebra Appl., 258 (1997), 311-343. doi: 10.1016/S0024-3795(96)00235-2.

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O. Gil-Medrano and P. W. Michor, The Riemannian manifold of all Riemannian metrics, Quart. J. of Math., 42 (1991), 183-202. doi: 10.1093/qmath/42.1.183.

[43]

G. H. Golub and H. A. van der Vorst, Eigenvalue computation in the 20th century, J. Comput. Appl. Math., 123 (2000), 35-65. doi: 10.1016/S0377-0427(00)00413-1.

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R. E. Greene and K. Shiohama, Diffeomorphisms and volume-preserving embeddings of noncompact manifolds, Trans. Amer. Math. Soc., 255 (1979), 403-414. doi: 10.1090/S0002-9947-1979-0542888-3.

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R. S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 65-222. doi: 10.1090/S0273-0979-1982-15004-2.

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U. Helmke and J. Moore, Singular-value decomposition via gradient and self-equivalent flows, Linear Algebra Appl., 169 (1992), 223-248. doi: 10.1016/0024-3795(92)90180-I.

[47]

U. HelmkeJ. Moore and J. Perkins, Dynamical systems that compute balanced realizations and the singular value decomposition, SIAM J. Matrix Anal. Appl., 15 (1994), 733-754. doi: 10.1137/S0895479891222490.

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R. Hermann, A sufficient condition that a mapping of Riemannian manifolds be a fibre bundle, Proc. Amer. Math. Soc., 11 (1960), 236-242. doi: 10.1090/S0002-9939-1960-0112151-4.

[49]

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Figure 1.  Illustration of the geometry of the polar decomposition of diffeomorphisms. The element $\nabla\phi$ in the factorization $\varphi=\nabla\phi\circ\psi$ is obtained at the intersection of the polar cone and the fiber of $\mu_1=\pi(\varphi)$. To compute $\nabla\phi$, one may start at $\varphi$ and follow a gradient flow constrained to the fiber of $\mu_1$ (vertical gradient flow, see $\S 2.2.1$), or one may take a gradient flow of a functional on the space of densities that approaches $\mu_1$ (entropy gradient flow, see $\S 2.2.2$) and lift it to a corresponding gradient flow on the polar cone (lifted gradient flow, see $\S 2.2.3$).
Figure 2.  Evolution of the matrix elements of $B(t)$ for the vertical gradient flow in Example 1. Notice that $B(0)=A$ and that $B(t)$ converges towards $P_{\infty}$ in (43) as $t\to\infty$.
Figure 3.  Convergence towards the limit $P_{\infty}$ of the vertical gradient flow in Example 1.
Figure 4.  Evolution of the lifted gradient flow in Example 2. Notice that $P(0)$ is the identity and that $P(t)$ converges towards $P_{\infty}$ in (54) as $t\to\infty$.
Figure 5.  Convergence towards the limit $P_{\infty}$ of the lifted gradient flow in Example 2. Notice that the convergence of both $-F(P(t))$ and $d^{2}(P(t),P_{\infty})$ as $t\to\infty$ is exponential, as fully explained by Theorem 2.14.
Figure 6.  Evolution of the lifted gradient flow in Example 3. Notice that $R(0)$ is the identity and that $R(t)$ converges towards $R_{\infty}$ in (80) as $t\to\infty$.
Figure 7.  Convergence towards the limit $R_{\infty}$ of the lifted gradient flow in Example 3. The convergence of both $-F(R(t))$ and $d^{2}(R(t),R_{\infty})$ is exponential with rate $\exp(-2t)$, as ensured by Theorem 3.10.
Figure 8.  Phase diagram of equation (86) for geodesics on ${\text{D}}(n)$. For every $l>0$ there is a unique integral curve $\lambda(t)$ such that $\lambda(0)=1$ and $\lambda(1)=l$. In consequence, every $\Lambda\in{\text{D}}(n)$ is connected to the identity $I$ by a unique horizontal geodesic.
Figure 9.  Evolution of the horizontal gradient flow in Example 4. $\Lambda(t)$ appears to converge to the ordered sequence of eigenvalues $(1/2,1,5)$.
Figure 10.  Convergence of $F(\Lambda(t))$ towards the minimum for the horizontal gradient flow in Example 4. The convergence appears to be exponential.
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