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Showing 1 to 20 of 33 for “"Wasserstein Distance"”.
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Distributionally robust binary classifier under Wasserstein distance
… with the radius calculated as per the Wasserstein distance. We derive the tractable formulation for the general problem. When focusing on the support vector machine (SVM), the general problem boils down to an easy-to-solve second- order cone programming problem. The robustified SVM is …
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Generative modeling using the sliced Wasserstein distance
… to improve stability, for instance, by using the Wasserstein distance rather than the Jenson-Shannon divergence. Here, we consider an alternative formulation for generative modeling based on random projections which, in its simplest form, results in a single objective rather than a saddlepoint …
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Non-parametric threshold for smoothed empirical Wasserstein distance
… distribution P. We show that when 𝐾 < 𝜎, the Wasserstein distance 𝑊₂² (Pₙ*𝒩(0, 𝜎² 𝐼 subscript 𝑑), P*𝒩 (0, 𝜎² 𝐼 subscript 𝑑)) converges at the parametric rate 𝑂(1/𝑛), and when 𝐾 > 𝜎, there exists a 𝐾-subgaussian distribution P such that 𝑊₂² (Pₙ *𝒩 (0, 𝜎² 𝐼 subscript 𝑑), P* 𝒩 (0, 𝜎² 𝐼 subscript …
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Deep generative models via explicit Wasserstein minimization
… target distributions, with the goal of a small Wasserstein distance (or other optimal transport costs). The approach is based on two principles: (a) if the source randomness of the network is a continuous distribution (the “semi-discrete” setting), then the Wasserstein distance is realized by a …
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Statistical problems in transport and alignment
… structure of this kind. First, we study the Wasserstein distance, a metric on the space of probability measures on an arbitrary metric space. We prove sharp rates of convergence for empirical measures in Wasserstein distance on sufficiently regular compact metric spaces, improving on a line …
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Robust Exact Algorithms for the Euclidean Bipartite Matching Problem
… of this problem is the computation of the p-Wasserstein distance which we define next. Given a complete bipartite graph with two disjoint sets of n points in d-dimensional Euclidean space and an integer p ≥ 1, let the cost of an edge be the p-th power of the Euclidean distance between its …
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Evaluating style modification in text
… judgment and a new set of automated metrics: the Wasserstein distance, word mover's distance on texts with style masked out, and adversarial classification for the respective aspects. Lastly, we illustrate aspect tradeoff curves for three state-of-the-art style transfer models to highlight the …
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On non-stationary Wishart matrices and functional Gaussian approximations in Hilbert spaces
… entries. We provide quantitative bounds in the Wasserstein distance for the cases of central convergence and non-central convergence, verify such convergences hold in the weak topology of C([a; b]; M_n(R)), and show that our result can be used to prove convergence in expectation of the empirical …
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Proximal Gradient Algorithms for Gaussian Variational Inference:Optimization in the Bures–Wasserstein Space
… a non-smooth term (the entropy) over the Bures–Wasserstein (BW) space of Gaussians endowed with the Wasserstein distance. For our proposed algorithm, we obtain state-of-the-art convergence guarantees when π is log-smooth and log-concave, as well as the first convergence guarantees to first-order …
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Topics in Deep Generative Modelling Mathematical and Computational Aspects of Diffusion Models and Generative Adversarial Networks
… We establish a theoretical upper bound on the Wasserstein 2-distance between distributions induced by stochastic and deterministic dynamics, linking it to the Fokker-Planck equation and its residual. Furthermore, the thesis explores the interplay between diffusion models and data manifolds. We …
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Sparse Expansion and Neuronal Disentanglement
… covering it. Interestingly, we show that the Wasserstein distance between a neuron’s output distribution and a Gaussian distribution is an indicator of its entanglement level and contribution to the accuracy of the model. Every layer of an LLM has highly entangled neurons, and model …
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Deterministic and Stochastic Approaches to Relaxation to Equilibrium for Particle Systems
… look at showing hypocoercivity in weak measure distances, and using probabilistic techniques. First we review the history of convergence to equilibrium for kinetic equations, particularly for spatially inhomogeneous kinetic theory (Boltzmann and Fokker-Planck equations) which motivates …
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Learning and inference with Wasserstein metrics
… tools from the study of optimal transport (or Wasserstein) distances between probability distributions. Optimal transport distances capture an intuitive notion of similarity between distributions, by incorporating the underlying geometry of the domain of the distributions. Despite their …
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Optimal transportation and barycenter problems via convex functions
… transportation problems and the associatedWasserstein distances in increasing levels of specificity. The first chapter introduces optimal transport problems and known properties characterizing optimal transport plans. The second chapter develops the Wasserstein distances, arising from …
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Scalable Combinatorial Algorithms for Optimal Transport Based Similarity Metrics
Optimal Transport (OT), also known as Wasserstein distance, is a valuable metric for comparing probability distributions. Owing to its appealing statistical properties, researchers in various fields, such as machine learning, use OT within applications. However, computing both exact and approximate …
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Machine Learning in Inverse Problems - Learning Regularisation Functionals and Operator Corrections
… images. The resulting regulariser decreases the Wasserstein distance between reconstructions and ground truth images at an optimal rate. We present computational results for computed tomography (CT) and magnetic resonance imaging (MRI) reconstruction and investigate generalisation properties of …
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Machine Learning Aided Aerial Radiation Mapping
… angular distribution of the incident radiation. Wasserstein distance is applied as a loss function to train the neural network for accurate prediction. Furthermore, radiation mapping could be enabled by performing directional measurements at different positions. In particular, optimization-based …
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Asymptotic Behaviour and Derivation of Mean Field Models
… and show contraction of its semi-group in the Wasserstein distance when the spatial variable lies on the torus. Chapter 9 studies a more complex model of passive transport of a solute under a large and highly oscillatory fluid field. We prove a homogenisation result showing convergence to an …
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Relaxation to equilibrium for kinetic Fokker-Planck equation
… e^{-C_2 t}\textrm{d}(f_0,\mu) \] for appropriate distances $\textit{d}$ and constants $C_1 \geq 1$, $C_2>0$. In Section 1 we provide an introduction and motivation for the equation, together with the setting of {Villani, Hypocoercivity} which will be useful in Section 2. In Section 2 we will …
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Two Random Multiplicative Processes: Multiplicative Cascades and Eigenvectors of the Random Schrodinger Operator
… it to compute the optimal Holder exponent in the Wasserstein distance on measures.In the second chapter, we focus on the eigenvectors of the one-dimensional discrete random Schrodinger operator. This is the Hamiltonian operator on the lattice of integers given by the discrete Laplacian plus an …
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