arXiv subjects

Zachariah Malik

Publications and source records attributed to Zachariah Malik.

5 recordsLinked to original sources

arXiv2026

Generative Modeling by Minimizing the Wasserstein-2 Loss

This paper develops a generative model by minimizing the second-order Wasserstein loss (the $W_2$ loss) through a distribution-dependent ordinary differential equation (ODE), whose dynamics involves the Kantorovich potential associated with the true data distribution and a current estimate of it. A main result shows that the time-marginal laws of the ODE form a gradient flow for the $W_2$ loss, which converges exponentially to the true data distribution. An Euler scheme for the ODE is proposed and it is shown to recover the gradient flow for the $W_2$ loss in the limit. An algorithm is designed by following the scheme and applying persistent training, which naturally fits our gradient-flow approach. In both low- and high-dimensional experiments, our algorithm outperforms Wasserstein generative adversarial networks by increasing the level of persistent training appropriately.

stat.ML↗

arXiv2026

Coherence Response in Noisy Quantum Measurements

Readout error models for noisy quantum devices almost universally assume that measurement noise is classical: the measurement statistics are obtained from the ideal computational-basis populations by a column-stochastic assignment matrix $A$. This description is equivalent to assuming that the effective positive-operator-valued measurement (POVM) is diagonal in the measurement basis, and therefore completely insensitive to quantum coherences. We relax this assumption and derive a fully general expression for the observed measurement probabilities under arbitrary completely positive trace-preserving (CPTP) noise preceding a computational-basis measurement. Writing the ideal post-circuit state $\tildeρ$ in terms of its populations $x$ and coherences $y$, we show that the observed probability vector $z$ satisfies $z = A x + C y$, where $A$ is the familiar classical assignment matrix and $C$ is a coherence-response matrix constructed from the off-diagonal matrix elements of the effective POVM in the computational basis. The classical model $z = A x$ arises if and only if all POVM elements are diagonal; in this sense $C$ quantifies accessible information about coherent readout distortions and interference between computational-basis states, all of which are invisible to models that retain only $A$. Our numerical experiments show that incorporating $C$ into readout recovery can improve fidelity over classical inversion and enable selective Pauli twirling with exponentially reduced circuit overhead. This work therefore provides a natural, fully general framework for coherence-sensitive readout modeling on current and future quantum devices.

quant-ph↗

arXiv2025

Mean-Field Langevin Diffusions with Density-dependent Temperature

In the context of non-convex optimization, we let the temperature of a Langevin diffusion to depend on the diffusion's own density function. The rationale is that the induced density captures to some extent the landscape imposed by the non-convex function to be minimized, such that a density-dependent temperature provides location-wise random perturbation that may better react to, for instance, the location and depth of local minimizers. As the Langevin dynamics is now self-regulated by its own density, it forms a mean-field stochastic differential equation (SDE) of the Nemytskii type, distinct from the standard McKean-Vlasov equations. Relying on Wasserstein subdifferential calculus, we first show that the corresponding (nonlinear) Fokker-Planck equation has a unique solution. Next, a weak solution to the SDE is constructed from the solution to the Fokker-Planck equation, by Trevisan's superposition principle. As time goes to infinity, we further show that the induced density converges to an invariant distribution, which admits an explicit formula in terms of the Lambert $W$ function. A numerical example suggests that the density-dependent temperature can simultaneously improve the accuracy of and rate of convergence to the estimate of global minimizers.

math.OC↗

arXiv2025

A Differential Equation Approach for Wasserstein GANs and Beyond

This paper proposes a new theoretical lens to view Wasserstein generative adversarial networks (WGANs). To minimize the Wasserstein-1 distance between the true data distribution and our estimate of it, we derive a distribution-dependent ordinary differential equation (ODE) which represents the gradient flow of the Wasserstein-1 loss, and show that a forward Euler discretization of the ODE converges. This inspires a new class of generative models that naturally integrates persistent training (which we call W1-FE). When persistent training is turned off, we prove that W1-FE reduces to WGAN. When we intensify persistent training, W1-FE is shown to outperform WGAN in training experiments from low to high dimensions, in terms of both convergence speed and training results. Intriguingly, one can reap the benefits only when persistent training is carefully integrated through our ODE perspective. As demonstrated numerically, a naive inclusion of persistent training in WGAN (without relying on our ODE framework) can significantly worsen training results.

stat.ML↗

arXiv2024

A competitive baseline for deep learning enhanced data assimilation using conditional Gaussian ensemble Kalman filtering

Ensemble Kalman Filtering (EnKF) is a popular technique for data assimilation, with far ranging applications. However, the vanilla EnKF framework is not well-defined when perturbations are nonlinear. We study two non-linear extensions of the vanilla EnKF - dubbed the conditional-Gaussian EnKF (CG-EnKF) and the normal score EnKF (NS-EnKF) - which sidestep assumptions of linearity by constructing the Kalman gain matrix with the `conditional Gaussian' update formula in place of the traditional one. We then compare these models against a state-of-the-art deep learning based particle filter called the score filter (SF). This model uses an expensive score diffusion model for estimating densities and also requires a strong assumption on the perturbation operator for validity. In our comparison, we find that CG-EnKF and NS-EnKF dramatically outperform SF for a canonical problem in high-dimensional multiscale data assimilation given by the Lorenz-96 system. Our analysis also demonstrates that the CG-EnKF and NS-EnKF can handle highly non-Gaussian additive noise perturbations, with the latter typically outperforming the former.

stat.ML↗