We address the problem of multichannel audio signal enhancement in reverberant environments with interfering sources. We propose an approach that leverages the Riemannian geometry of the spatial correlation matrices of the received signals to estimate the relative transfer function (RTF) of the desired source. Specifically, we compute the spatial correlation matrices in short-time segments, and subsequently, their Riemannian mean, which preserves shared spatial components while attenuating unshared ones. This enables an effective rejection of intermittent interference, leading to accurate RTF estimation. We experimentally show that when the proposed RTF estimation is incorporated into the Minimum Variance Distortionless Response (MVDR) beamformer, it enhances the desired signal, outperforming the MVDR beamformer that is based on standard (Euclidean) RTF estimation. These favorable experimental results are demonstrated in challenging acoustic environments including multiple strong interfering sources, noise, and reverberations.

Langevin dynamics (LD) is widely used for sampling from distributions and for optimization. In this work, we derive a closed-form expression for the expected loss of preconditioned LD near stationary points of the objective function. We use the fact that at the vicinity of such points, LD reduces to an Ornstein–Uhlenbeck process, which is amenable to convenient mathematical treatment. Our analysis reveals that when the preconditioning matrix satisfies a particular relation with respect to the noise covariance, LD's expected loss becomes proportional to the rank of the objective's Hessian. We illustrate the applicability of this result in the context of neural networks, where the Hessian rank has been shown to capture the complexity of the predictor function but is usually computationally hard to probe. Finally, we use our analysis to compare SGD-like and Adam-like preconditioners and identify the regimes under which each of them leads to a lower expected loss.

Direct Position Determination (DPD) is the state-of-the-art solution for emitter localization using multiple phased arrays. This paper shows that DPD can be recast as a covariance-fitting (CF) problem that minimizes the Euclidean distance between a sample covariance matrix ${\mathbf{\hat R}}$ and its location-dependent model R. By showing equivalence to existing DPD methods, this CF viewpoint highlights that the geometry of the Hermitian Positive Definite (HPD) covariance matrices R and ${\mathbf{\hat R}}$ is simply overlooked. Based on this critical observation, we propose a new CF approach for DPD that specifically exploits the Riemannian geometry of HPD matrices for measuring the distance between R and ${\mathbf{\hat R}}$. Experimental results showcase that the proposed Riemannian CF approach for DPD leads to a significant improvement in localization accuracy.

Options have been shown to be an effective tool in reinforcement learning, facilitating improved exploration and learning. In this paper, we present an approach based on spectral graph theory and derive an algorithm that systematically discovers options without access to a specific reward or task assignment. As opposed to the common practice used in previous methods, our algorithm makes full use of the spectrum of the graph Laplacian. Incorporating modes associated with higher graph frequencies unravels domain subtleties, which are shown to be useful for option discovery. Using geometric and manifold-based analysis, we present a theoretical justification for the algorithm. In addition, we showcase its performance in several domains, demonstrating clear improvements compared to competing methods.