The goal of Bayesian deep learning is to provide uncertainty quantification via the posterior distribution. However, exact inference over the weight space is computationally intractable due to the ultra-high dimensions of the neural network. Variational inference (VI) is a promising approach, but naive application on weight space does not scale well and often underperform on predictive accuracy. In this paper, we propose a new adaptive variational Bayesian algorithm to train neural networks on weight space that achieves high predictive accuracy. By showing that there is an equivalence to Stochastic Gradient Hamiltonian Monte Carlo(SGHMC) with preconditioning matrix, we then propose an MCMC within EM algorithm, which incorporates the spike-and-slab prior to capture the sparsity of the neural network. The EM-MCMC algorithm allows us to perform optimization and model pruning within one-shot. We evaluate our methods on CIFAR-10, CIFAR-100 and ImageNet datasets, and demonstrate that our dense model can reach the state-of-the-art performance and our sparse model perform very well compared to previously proposed pruning schemes.

In this paper, we focus on the intersections of a manifold's local structures to analyze the global structure of a manifold. We obtain local regions on data manifolds such as the latent space of StyleGAN2, using Mapper, a tool from topological data analysis. We impose gluing compatibility conditions on overlapping local regions, which guarantee that the local structures can be glued together to the global structure of a manifold. We propose a novel generative flow model called Atlas flow that uses compatibility to reattach the local regions. Our model shows that the generating processes perform well on synthetic dataset samples of well-known manifolds with noise. Furthermore, we investigate the style vector manifold of StyleGAN2 using our model.

Dataset distillation aims to learn a small synthetic dataset that preserves most of the information from the original dataset. Dataset distillation can be formulated as a bi-level meta-learning problem where the outer loop optimizes the meta-dataset and the inner loop trains a model on the distilled data. Meta-gradient computation is one of the key challenges in this formulation, as differentiating through the inner loop learning procedure introduces significant computation and memory costs. In this paper, we address these challenges using neural Feature Regression with Pooling (FRePo), achieving the state-of-the-art performance with an order of magnitude less memory requirement and two orders of magnitude faster training than previous methods. The proposed algorithm is analogous to truncated backpropagation through time with a pool of models to alleviate various types of overfitting in dataset distillation. FRePo significantly outperforms the previous methods on CIFAR100, Tiny ImageNet, and ImageNet-1K. Furthermore, we show that high-quality distilled data can greatly improve various downstream applications, such as continual learning and membership inference defense. Please check out our webpage at https://sites.google.com/view/frepo.

Dimensionality reduction techniques aim at representing high-dimensional data in low-dimensional spaces to extract hidden and useful information or facilitate visual understanding and interpretation of the data. However, few of them take into consideration the potential cluster information contained implicitly in the high-dimensional data. In this paper, we propose LaptSNE, a new graph-layout nonlinear dimensionality reduction method based on t-SNE, one of the best techniques for visualizing high-dimensional data as 2D scatter plots. Specifically, LaptSNE leverages the eigenvalue information of the graph Laplacian to shrink the potential clusters in the low-dimensional embedding when learning to preserve the local and global structure from high-dimensional space to low-dimensional space. It is nontrivial to solve the proposed model because the eigenvalues of normalized symmetric Laplacian are functions of the decision variable. We provide a majorization-minimization algorithm with convergence guarantee to solve the optimization problem of LaptSNE and show how to calculate the gradient analytically, which may be of broad interest when considering optimization with Laplacian-composited objective. We evaluate our method by a formal comparison with state-of-the-art methods on seven benchmark datasets, both visually and via established quantitative measurements. The results demonstrate the superiority of our method over baselines such as t-SNE and UMAP. We also provide out-of-sample extension, large-scale extension and mini-batch extension for our LaptSNE to facilitate dimensionality reduction in various scenarios.

Quantum machine learning promises to efficiently solve important problems. There are two persistent challenges in classical machine learning: the lack of labeled data, and the limit of computational power. We propose a novel framework that resolves both issues: quantum semi-supervised learning. Moreover, we provide a protocol in systematically designing quantum machine learning algorithms with quantum supremacy, which can be extended beyond quantum semi-supervised learning. In the meantime, we show that naive quantum matrix product estimation algorithm outperforms the best known classical matrix multiplication algorithm. We showcase two concrete quantum semi-supervised learning algorithms: a quantum self-training algorithm named the propagating nearest-neighbor classifier, and the quantum semi-supervised K-means clustering algorithm. By doing time complexity analysis, we conclude that they indeed possess quantum supremacy.

Open-Set Domain Adaptation (OSDA) assumes that a target domain contains unknown classes, which are not discovered in a source domain. Existing domain adversarial learning methods are not suitable for OSDA because distribution matching with $\textit{unknown}$ classes leads to negative transfer. Previous OSDA methods have focused on matching the source and the target distribution by only utilizing $\textit{known}$ classes. However, this $\textit{known}$-only matching may fail to learn the target-$\textit{unknown}$ feature space. Therefore, we propose Unknown-Aware Domain Adversarial Learning (UADAL), which $\textit{aligns}$ the source and the target-$\textit{known}$ distribution while simultaneously $\textit{segregating}$ the target-$\textit{unknown}$ distribution in the feature alignment procedure. We provide theoretical analyses on the optimized state of the proposed $\textit{unknown-aware}$ feature alignment, so we can guarantee both $\textit{alignment}$ and $\textit{segregation}$ theoretically. Empirically, we evaluate UADAL on the benchmark datasets, which shows that UADAL outperforms other methods with better feature alignments by reporting state-of-the-art performances.

Principal component analysis (PCA) is a dimensionality reduction method in data analysis that involves diagonalizing the covariance matrix of the dataset. Recently, quantum algorithms have been formulated for PCA based on diagonalizing a density matrix. These algorithms assume that the covariance matrix can be encoded in a density matrix, but a concrete protocol for this encoding has been lacking. Our work aims to address this gap. Assuming amplitude encoding of the data, with the data given by the ensemble $\{p_i,| \psi_i \rangle\}$, then one can easily prepare the ensemble average density matrix $\overline{\rho} = \sum_i p_i |\psi_i\rangle \langle \psi_i |$. We first show that $\overline{\rho}$ is precisely the covariance matrix whenever the dataset is centered. For quantum datasets, we exploit global phase symmetry to argue that there always exists a centered dataset consistent with $\overline{\rho}$, and hence $\overline{\rho}$ can always be interpreted as a covariance matrix. This provides a simple means for preparing the covariance matrix for arbitrary quantum datasets or centered classical datasets. For uncentered classical datasets, our method is so-called "PCA without centering", which we interpret as PCA on a symmetrized dataset. We argue that this closely corresponds to standard PCA, and we derive equations and inequalities that bound the deviation of the spectrum obtained with our method from that of standard PCA. We numerically illustrate our method for the MNIST handwritten digit dataset. We also argue that PCA on quantum datasets is natural and meaningful, and we numerically implement our method for molecular ground-state datasets.

In this paper, we establish the global optimality and convergence rate of an off-policy actor critic algorithm in the tabular setting without using density ratio to correct the discrepancy between the state distribution of the behavior policy and that of the target policy. Our work goes beyond existing works on the optimality of policy gradient methods in that existing works use the exact policy gradient for updating the policy parameters while we use an approximate and stochastic update step. Our update step is not a gradient update because we do not use a density ratio to correct the state distribution, which aligns well with what practitioners do. Our update is approximate because we use a learned critic instead of the true value function. Our update is stochastic because at each step the update is done for only the current state action pair. Moreover, we remove several restrictive assumptions from existing works in our analysis. Central to our work is the finite sample analysis of a generic stochastic approximation algorithm with time-inhomogeneous update operators on time-inhomogeneous Markov chains, based on its uniform contraction properties.

Fitting a polynomial to observed data is an ubiquitous task in many signal processing and machine learning tasks, such as interpolation and prediction. In that context, input and output pairs are available and the goal is to find the coefficients of the polynomial. However, in many applications, the input may be partially known or not known at all, rendering conventional regression approaches not applicable. In this paper, we formally state the (potentially partial) blind regression problem, illustrate some of its theoretical properties, and propose algorithmic approaches to solve it. As a case-study, we apply our methods to a jitter-correction problem and corroborate its performance.

Recognizing an image and segmenting it into coherent regions are often treated as separate tasks. Human vision, however, has a general sense of segmentation hierarchy before recognition occurs. We are thus inspired to learn image recognition with hierarchical image segmentation based entirely on unlabeled images. Our insight is to learn fine-to-coarse features concurrently at superpixels, segments, and full image levels, enforcing consistency and goodness of feature induced segmentations while maximizing discrimination among image instances. Our model innovates vision transformers on three aspects. 1) We use adaptive segment tokens instead of fixed-shape patch tokens. 2) We create a token hierarchy by inserting graph pooling between transformer blocks, naturally producing consistent multi-scale segmentations while increasing the segment size and reducing the number of tokens. 3) We produce hierarchical image segmentation for free while training for recognition by maximizing image-wise discrimination. Our work delivers the first concurrent recognition and hierarchical segmentation model without any supervision. Validated on ImageNet and PASCAL VOC, it achieves better recognition and segmentation with higher computational efficiency.