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

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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. Google Scholar

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M. T. Chu, A list of matrix flows with applications, Fields Institute Communications, 3 (1994), 87-97. Google Scholar

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

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

[28]

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[31]

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

[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. Google Scholar

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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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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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. Google Scholar

[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. Google Scholar

[38]

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

[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. Google Scholar

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

[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. Google Scholar

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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. Google Scholar

[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. Google Scholar

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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. Google Scholar

[45]

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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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