arXiv subjects

Ting-Kam Leonard Wong

Publications and source records attributed to Ting-Kam Leonard Wong.

33 records · Page 2Linked to original sources

arXiv2021

Pseudo-Riemannian geometry embeds information geometry in optimal transport

Optimal transport and information geometry both study geometric structures on spaces of probability distributions. Optimal transport characterizes the cost-minimizing movement from one distribution to another, while information geometry originates from coordinate-invariant properties of statistical inference. Their connections and applications in statistics and machine learning have started to gain more attention. In this paper we give a new differential geometric connection between the two fields. Namely, the pseudo-Riemannian framework of Kim and McCann, a geometric perspective on the fundamental Ma-Trudinger-Wang (MTW) condition in the regularity theory of optimal transport maps, encodes the dualistic structure of statistical manifold. This general relation is described using the natural framework of $c$-divergence, a divergence defined by an optimal transport map. As a by-product, we obtain a new information-geometric interpretation of the MTW tensor. This connection sheds light on old and new aspects of information geometry. The dually flat geometry of Bregman divergence corresponds to the quadratic cost and the pseudo-Euclidean space, and the $L^{(α)}$-divergence introduced by Pal and the first author has constant sectional curvature in a sense to be made precise. In these cases we give a geometric interpretation of the information-geometric curvature in terms of the divergence between a primal-dual pair of geodesics.

math.DG↗

arXiv2020

Scalable Gradients for Stochastic Differential Equations

The adjoint sensitivity method scalably computes gradients of solutions to ordinary differential equations. We generalize this method to stochastic differential equations, allowing time-efficient and constant-memory computation of gradients with high-order adaptive solvers. Specifically, we derive a stochastic differential equation whose solution is the gradient, a memory-efficient algorithm for caching noise, and conditions under which numerical solutions converge. In addition, we combine our method with gradient-based stochastic variational inference for latent stochastic differential equations. We use our method to fit stochastic dynamics defined by neural networks, achieving competitive performance on a 50-dimensional motion capture dataset.

cs.LG↗

arXiv2020

Multiplicative Schrödinger problem and the Dirichlet transport

We consider an optimal transport problem on the unit simplex whose solutions are given by gradients of exponentially concave functions and prove two main results. First, we show that the optimal transport is the large deviation limit of a particle system of Dirichlet processes transporting one probability measure on the unit simplex to another by coordinatewise multiplication and normalizing. The structure of our Lagrangian and the appearance of the Dirichlet process relate our problem closely to the entropic measure on the Wasserstein space as defined by von-Renesse and Sturm in the context of Wasserstein diffusion. The limiting procedure is a triangular limit where we allow simultaneously the number of particles to grow to infinity while the `noise' tends to zero. The method, which generalizes easily to many other cost functions, including the squared Euclidean distance, provides a novel combination of the Schrödinger problem approach due to C. Léonard and the related Brownian particle systems by Adams et al.which does not require gamma convergence. Second, we analyze the behavior of entropy along the paths of transport. The reference measure on the simplex is taken to be the Dirichlet measure with all zero parameters which relates to the finite-dimensional distributions of the entropic measure. The interpolating curves are not the usual McCann lines. Nevertheless we show that entropy plus a multiple of the transport cost remains convex, which is reminiscent of the semiconvexity of entropy along lines of McCann interpolations in negative curvature spaces. We also obtain, under suitable conditions, dimension-free bounds of the optimal transport cost in terms of entropy.

math.PR↗

arXiv2020

Time-consistent conditional expectation under probability distortion

We introduce a new notion of conditional nonlinear expectation under probability distortion. Such a distorted nonlinear expectation is not sub-additive in general, so it is beyond the scope of Peng's framework of nonlinear expectations. A more fundamental problem when extending the distorted expectation to a dynamic setting is time-inconsistency, that is, the usual "tower property" fails. By localizing the probability distortion and restricting to a smaller class of random variables, we introduce a so-called distorted probability and construct a conditional expectation in such a way that it coincides with the original nonlinear expectation at time zero, but has a time-consistent dynamics in the sense that the tower property remains valid. Furthermore, we show that in the continuous time model this conditional expectation corresponds to a parabolic differential equation whose coefficient involves the law of the underlying diffusion. This work is the first step towards a new understanding of nonlinear expectations under probability distortion, and will potentially be a helpful tool for solving time-inconsistent stochastic optimization problems.

q-fin.MF↗

arXiv2019

Logarithmic divergences: geometry and interpretation of curvature

We study the logarithmic $L^{(α)}$-divergence which extrapolates the Bregman divergence and corresponds to solutions to novel optimal transport problems. We show that this logarithmic divergence is equivalent to a conformal transformation of the Bregman divergence, and, via an explicit affine immersion, is equivalent to Kurose's geometric divergence. In particular, the $L^{(α)}$-divergence is a canonical divergence of a statistical manifold with constant sectional curvature $-α$. For such a manifold, we give a geometric interpretation of its sectional curvature in terms of how the divergence between a pair of primal and dual geodesics differ from the dually flat case. Further results can be found in our follow-up paper [27] which uncovers a novel relation between optimal transport and information geometry.

math.DG↗

arXiv2018

Logarithmic divergences from optimal transport and Rényi geometry

Divergences, also known as contrast functions, are distance-like quantities defined on manifolds of non-negative or probability measures. Using the duality in optimal transport, we introduce and study the one-parameter family of $L^{(\pm α)}$-divergences. It includes the Bregman divergence corresponding to the Euclidean quadratic cost, and the $L$-divergence introduced by Pal and the author in connection with portfolio theory and a logarithmic cost function. They admit natural generalizations of exponential family that are closely related to the $α$-family and $q$-exponential family. In particular, the $L^{(\pm α)}$-divergences of the corresponding potential functions are Rényi divergences. Using this unified framework we prove that the induced geometries are dually projectively flat with constant sectional curvatures, and a generalized Pythagorean theorem holds true. Conversely, we show that if a statistical manifold is dually projectively flat with constant curvature $\pm α$ with $α> 0$, then it is locally induced by an $L^{(\mp α)}$-divergence. We define in this context a canonical divergence which extends the one for dually flat manifolds.

math.PR↗

arXiv2018

Random walks and induced Dirichlet forms on compact spaces of homogeneous type

We extend our study of random walks and induced Dirichlet forms on self-similar sets [arXiv:1604.05440, 1612.01708] to compact spaces of homogeneous type $(K, ρ,μ)$. A successive partition on $K$ brings a natural augmented tree structure $(X, E)$ that is Gromov hyperbolic, and the hyperbolic boundary is Hölder equivalent to $K$. We then introduce a class of transient reversible random walks on $(X, E)$ with return ratio $λ$. Using Silverstein's theory of Markov chains, we prove that the random walk induces an energy form on $K$ with $$ {\mathcal E}_K [u] \asymp \iint_{K\times K \setminus Δ} \frac{|u(ξ) - u(η)|^2}{V(ξ, η)ρ(ξ, η)^β} dμ(ξ) dμ(η), $$ where $V(ξ, η)$ is the $μ$-volume of the ball centered at $ξ$ with radius $ρ(ξ, η)$, $Δ$ is the diagonal, and $β$ depends on $λ$. In particular, for an $α$-set in ${\mathbb R}^d$, the kernel of the energy form is of order $\frac{1}{|ξ-η|^{α+β}}$. We also discuss conditions for this energy form to be a non-local regular Dirichlet form.

math.PR↗

arXiv2017

Random walks and induced Dirichlet forms on self-similar sets

Let $K$ be a self-similar set satisfying the open set condition. Following Kaimanovich's elegant idea, it has been proved that on the symbolic space $X$ of $K$ a natural augmented tree structure ${\mathfrak E}$ exists; it is hyperbolic, and the hyperbolic boundary $\partial_HX$ with the Gromov metric is Hölder equivalent to $K$. In this paper we consider certain reversible random walks with return ratio $0< λ<1$ on $(X, {\mathfrak E})$. We show that the Martin boundary ${\mathcal M}$ can be identified with $\partial_H X$ and $K$. With this setup and a device of Silverstein, we obtain precise estimates of the Martin kernel and the Naïm kernel in terms of the Gromov product. Moreover, the Naïm kernel turns out to be a jump kernel satisfying the estimate $Θ(ξ, η) \asymp |ξ-η|^{-(α+ β)}$, where $α$ is the Hausdorff dimension of $K$ and $β$ depends on $λ$. For suitable $β$, the kernel defines a regular non-local Dirichlet form on $K$. This extends the results of Kigami concerning random walks on certain trees with Cantor-type sets as boundaries.

math.PR↗

arXiv2017

On portfolios generated by optimal transport

First introduced by Fernholz in stochastic portfolio theory, functionally generated portfolio allows its investment performance to be attributed to directly observable and easily interpretable market quantities. In previous works we showed that Fernholz's multiplicatively generated portfolio has deep connections with optimal transport and the information geometry of exponentially concave functions. Recently, Karatzas and Ruf introduced a new additive portfolio generation whose relation with optimal transport was studied by Vervuurt. We show that additively generated portfolio can be interpreted in terms of the well-known dually flat information geometry of Bregman divergence. Moreover, we characterize, in a sense to be made precise, all possible forms of functional portfolio constructions that contain additive and multiplicative generations as special cases. Each construction involves a divergence functional on the unit simplex measuring the market volatility captured, and admits a pathwise decomposition for the portfolio value. We illustrate with an empirical example.

q-fin.MF↗

arXiv2017

Exponentially concave functions and a new information geometry

A function is exponentially concave if its exponential is concave. We consider exponentially concave functions on the unit simplex. In a previous paper we showed that gradient maps of exponentially concave functions provide solutions to a Monge-Kantorovich optimal transport problem and give a better gradient approximation than those of ordinary concave functions. The approximation error, called L-divergence, is different from the usual Bregman divergence. Using tools of information geometry and optimal transport, we show that L-divergence induces a new information geometry on the simplex consisting of a Riemannian metric and a pair of dually coupled affine connections which defines two kinds of geodesics. We show that the induced geometry is dually projectively flat but not flat. Nevertheless, we prove an analogue of the celebrated generalized Pythagorean theorem from classical information geometry. On the other hand, we consider displacement interpolation under a Lagrangian integral action that is consistent with the optimal transport problem and show that the action minimizing curves are dual geodesics. The Pythagorean theorem is also shown to have an interesting application of determining the optimal trading frequency in stochastic portfolio theory.

math.PR↗

arXiv2016

Universal portfolios in stochastic portfolio theory

Consider a family of portfolio strategies with the aim of achieving the asymptotic growth rate of the best one. The idea behind Cover's universal portfolio is to build a wealth-weighted average which can be viewed as a buy-and-hold portfolio of portfolios. When an optimal portfolio exists, the wealth-weighted average converges to it by concentration of wealth. Working under a discrete time and pathwise setup, we show under suitable conditions that the distribution of wealth in the family satisfies a pathwise large deviation principle as time tends to infinity. Our main result extends Cover's portfolio to the nonparametric family of functionally generated portfolios in stochastic portfolio theory and establishes its asymptotic universality.

q-fin.PM↗

arXiv2016

Cover's universal portfolio, stochastic portfolio theory and the numeraire portfolio

Cover's celebrated theorem states that the long run yield of a properly chosen "universal" portfolio is as good as the long run yield of the best retrospectively chosen constant rebalanced portfolio. The "universality" pertains to the fact that this result is model-free, i.e., not dependent on an underlying stochastic process. We extend Cover's theorem to the setting of stochastic portfolio theory as initiated by R. Fernholz: the rebalancing rule need not to be constant anymore but may depend on the present state of the stock market. This model-free result is complemented by a comparison with the log-optimal numeraire portfolio when fixing a stochastic model of the stock market. Roughly speaking, under appropriate assumptions, the optimal long run yield coincides for the three approaches mentioned in the title of this paper. We present our results in discrete and continuous time.

q-fin.MF↗

arXiv2016

Energy, entropy, and arbitrage

We introduce a pathwise approach to analyze the relative performance of an equity portfolio with respect to a benchmark market portfolio. In this energy-entropy framework, the relative performance is decomposed into three components: a volatility term, a relative entropy term measuring the distance between the portfolio weights and the market capital distribution, and another entropy term that can be controlled by the investor by adopting a suitable rebalancing strategy. This framework leads to a class of portfolio strategies that allows one to outperform, in the long run, a market that is diverse and sufficiently volatile in the sense of stochastic portfolio theory. The framework is illustrated with several empirical examples.

q-fin.PM↗

arXiv2015

The geometry of relative arbitrage

Consider an equity market with $n$ stocks. The vector of proportions of the total market capitalizations that belong to each stock is called the market weight. The market weight defines the market portfolio which is a buy-and-hold portfolio representing the performance of the entire stock market. Consider a function that assigns a portfolio vector to each possible value of the market weight, and we perform self-financing trading using this portfolio function. We study the problem of characterizing functions such that the resulting portfolio will outperform the market portfolio in the long run under the conditions of diversity and sufficient volatility. No other assumption on the future behavior of stock prices is made. We prove that the only solutions are functionally generated portfolios in the sense of Fernholz. A second characterization is given as the optimal maps of a remarkable optimal transport problem. Both characterizations follow from a novel property of portfolios called multiplicative cyclical monotonicity.

q-fin.PM↗

arXiv2014

Optimization of relative arbitrage

In stochastic portfolio theory, a relative arbitrage is an equity portfolio which is guaranteed to outperform a benchmark portfolio over a finite horizon. When the market is diverse and sufficiently volatile, and the benchmark is the market or a buy-and-hold portfolio, functionally generated portfolios introduced by Fernholz provide a systematic way of constructing relative arbitrages. In this paper we show that if the market portfolio is replaced by the equal or entropy weighted portfolio among many others, no relative arbitrages can be constructed under the same conditions using functionally generated portfolios. We also introduce and study a shaped-constrained optimization problem for functionally generated portfolios in the spirit of maximum likelihood estimation of a log-concave density.

q-fin.PM↗