Spectral clustering has gained popularity for clustering non-convex data due to its simplicity and effectiveness. It is essential to construct a similarity graph using a high-quality affinity measure that models the local neighborhood relations among the data samples. However, incomplete data can lead to inaccurate affinity measures, resulting in degraded clustering performance. To address these issues, we propose an imputation-free framework with two novel approaches to improve spectral clustering on incomplete data. Firstly, we introduce a new kernel correction method that enhances the quality of the kernel matrix estimated on incomplete data with a theoretical guarantee, benefiting classical spectral clustering on pre-defined kernels.

Graph-based semi-supervised learning (GSSL) serves as a powerful tool to model the underlying manifold structures of samples in high-dimensional spaces. It involves two phases: constructing an affinity graph from available data and inferring labels for unlabeled nodes on this graph. While numerous algorithms have been developed for label inference, the crucial graph construction phase has received comparatively less attention, despite its significant influence on the subsequent phase. In this paper, we present an optimal asymmetric graph structure for the label inference phase with theoretical motivations. Unlike existing graph construction methods, we differentiate the distinct roles that labeled nodes and unlabeled nodes could play.

Remote photoplethysmography (rPPG) enables non-invasive extraction of blood volume pulse signals through imaging, transforming spatial-temporal data into time series signals. Advances in end-to-end rPPG approaches have focused on this transformation where attention mechanisms are crucial for feature extraction. However, existing methods compute attention disjointly across spatial, temporal, and channel dimensions. Here, we propose the Factorized Self-Attention Module (FSAM), which jointly computes multidimensional attention from voxel embeddings using nonnegative matrix factorization. To demonstrate FSAM's effectiveness, we developed FactorizePhys, an end-to-end 3D-CNN architecture for estimating blood volume pulse signals from raw video frames.

Geometric deep learning enables the encoding of physical symmetries in modeling 3D objects.

Solving the quantum many-body Schrödinger equation is a fundamental and challenging problem in the fields of quantum physics, quantum chemistry, and material sciences. One of the common computational approaches to this problem is Quantum Variational Monte Carlo (QVMC), in which ground-state solutions are obtained by minimizing the energy of the system within a restricted family of parameterized wave functions. Deep learning methods partially address the limitations of traditional QVMC by representing a rich family of wave functions in terms of neural networks. However, the optimization objective in QVMC remains notoriously hard to minimize and requires second-order optimization methods such as natural gradient. In this paper, we first reformulate energy functional minimization in the space of Born distributions corresponding to particle-permutation (anti-)symmetric wave functions, rather than the space of wave functions. We then interpret QVMC as the Fisher-Rao gradient flow in this distributional space, followed by a projection step onto the variational manifold. This perspective provides us with a principled framework to derive new QMC algorithms, by endowing the distributional space with better metrics, and following the projected gradient flow induced by those metrics. More specifically, we propose "Wasserstein Quantum Monte Carlo" (WQMC), which uses the gradient flow induced by the Wasserstein metric, rather than the Fisher-Rao metric, and corresponds to transporting the probability mass, rather than teleporting it. We demonstrate empirically that the dynamics of WQMC results in faster convergence to the ground state of molecular systems.