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We study the spectral convergence of a symmetrized Graph Laplacian matrix induced by a Gaussian kernel evaluated on pairs of embedded data, sampled from a manifold with boundary, a sub-manifold of $\mathbb{R}^m$. Specifically, we deduce the convergence rates for eigenpairs of the discrete Graph-Laplacian matrix to the eigensolutions of the Laplace-Beltrami operator that are well-defined on manifolds with boundary, including the homogeneous Neumann and Dirichlet boundary conditions. For the Dirichlet problem, we deduce the convergence of the \emph{truncated Graph Laplacian}, which is recently numerically observed in applications, and provide a detailed numerical investigation on simple manifolds. Our method of proof relies on the min-max argument over a compact and symmetric integral operator, leveraging the RKHS theory for spectral convergence of integral operator and a recent pointwise asymptotic result of a Gaussian kernel integral operator on manifolds with boundary.

In recent years, significant attention in deep learning theory has been devoted to analyzing the generalization performance of models with multiple layers of Gaussian random features. However, few works have considered the effect of feature anisotropy; most assume that features are generated using independent and identically distributed Gaussian weights. Here, we derive learning curves for models with many layers of structured Gaussian features. We show that allowing correlations between the rows of the first layer of features can aid generalization, while structure in later layers is generally detrimental. Our results shed light on how weight structure affects generalization in a simple class of solvable models.

We compute precise asymptotic expressions for the learning curves of least squares random feature (RF) models with either a separable strongly convex regularization or the $\ell_1$ regularization. We propose a novel multi-level application of the convex Gaussian min max theorem (CGMT) to overcome the traditional difficulty of finding computable expressions for random features models with correlated data. Our result takes the form of a computable 4-dimensional scalar optimization. In contrast to previous results, our approach does not require solving an often intractable proximal operator, which scales with the number of model parameters. Furthermore, we extend the universality results for the training and generalization errors for RF models to $\ell_1$ regularization. In particular, we demonstrate that under mild conditions, random feature models with elastic net or $\ell_1$ regularization are asymptotically equivalent to a surrogate Gaussian model with the same first and second moments. We numerically demonstrate the predictive capacity of our results, and show experimentally that the predicted test error is accurate even in the non-asymptotic regime.

We present Bit Diffusion: a simple and generic approach for generating discrete data with continuous state and continuous time diffusion models. The main idea behind our approach is to first represent the discrete data as binary bits, and then train a continuous diffusion model to model these bits as real numbers which we call analog bits. To generate samples, the model first generates the analog bits, which are then thresholded to obtain the bits that represent the discrete variables. We further propose two simple techniques, namely Self-Conditioning and Asymmetric Time Intervals, which lead to a significant improvement in sample quality. Despite its simplicity, the proposed approach can achieve strong performance in both discrete image generation and image captioning tasks. For discrete image generation, we significantly improve previous state-of-the-art on both CIFAR-10 (which has 3K discrete 8-bit tokens) and ImageNet-64x64 (which has 12K discrete 8-bit tokens), outperforming the best autoregressive model in both sample quality (measured by FID) and efficiency. For image captioning on MS-COCO dataset, our approach achieves competitive results compared to autoregressive models.

Incomplete covariate vectors are known to be problematic for estimation and inferences on model parameters, but their impact on prediction performance is less understood. We develop an imputation-free method that builds on a random partition model admitting variable-dimension covariates. Cluster-specific response models further incorporate covariates via linear predictors, facilitating estimation of smooth prediction surfaces with relatively few clusters. We exploit marginalization techniques of Gaussian kernels to analytically project response distributions according to any pattern of missing covariates, yielding a local regression with internally consistent uncertainty propagation that utilizes only one set of coefficients per cluster. Aggressive shrinkage of these coefficients regulates uncertainty due to missing covariates. The method allows in- and out-of-sample prediction for any missingness pattern, even if the pattern in a new subject's incomplete covariate vector was not seen in the training data. We develop an MCMC algorithm for posterior sampling that improves a computationally expensive update for latent cluster allocation. Finally, we demonstrate the model's effectiveness for nonlinear point and density prediction under various circumstances by comparing with other recent methods for regression of variable dimensions on synthetic and real data.

Many problems arising in control require the determination of a mathematical model of the application. This has often to be performed starting from input-output data, leading to a task known as system identification in the engineering literature. One emerging topic in this field is estimation of networks consisting of several interconnected dynamic systems. We consider the linear setting assuming that system outputs are the result of many correlated inputs, hence making system identification severely ill-conditioned. This is a scenario often encountered when modeling complex cybernetics systems composed by many sub-units with feedback and algebraic loops. We develop a strategy cast in a Bayesian regularization framework where any impulse response is seen as realization of a zero-mean Gaussian process. Any covariance is defined by the so called stable spline kernel which includes information on smooth exponential decay. We design a novel Markov chain Monte Carlo scheme able to reconstruct the impulse responses posterior by efficiently dealing with collinearity. Our scheme relies on a variation of the Gibbs sampling technique: beyond considering blocks forming a partition of the parameter space, some other (overlapping) blocks are also updated on the basis of the level of collinearity of the system inputs. Theoretical properties of the algorithm are studied obtaining its convergence rate. Numerical experiments are included using systems containing hundreds of impulse responses and highly correlated inputs.

We consider random sample splitting for estimation and inference in high dimensional generalized linear models, where we first apply the lasso to select a submodel using one subsample and then apply the debiased lasso to fit the selected model using the remaining subsample. We show that, no matter including a prespecified subset of regression coefficients or not, the debiased lasso estimation of the selected submodel after a single splitting follows a normal distribution asymptotically. Furthermore, for a set of prespecified regression coefficients, we show that a multiple splitting procedure based on the debiased lasso can address the loss of efficiency associated with sample splitting and produce asymptotically normal estimates under mild conditions. Our simulation results indicate that using the debiased lasso instead of the standard maximum likelihood estimator in the estimation stage can vastly reduce the bias and variance of the resulting estimates. We illustrate the proposed multiple splitting debiased lasso method with an analysis of the smoking data of the Mid-South Tobacco Case-Control Study.

Previous research has shown that established techniques for spoken voice conversion (VC) do not perform as well when applied to singing voice conversion (SVC). We propose an alternative loss component in a loss function that is otherwise well-established among VC tasks, which has been shown to improve our model's SVC performance. We first trained a singer identity embedding (SIE) network on mel-spectrograms of singer recordings to produce singer-specific variance encodings using contrastive learning. We subsequently trained a well-known autoencoder framework (AutoVC) conditioned on these SIEs, and measured differences in SVC performance when using different latent regressor loss components. We found that using this loss w.r.t. SIEs leads to better performance than w.r.t. bottleneck embeddings, where converted audio is more natural and specific towards target singers. The inclusion of this loss component has the advantage of explicitly forcing the network to reconstruct with timbral similarity, and also negates the effect of poor disentanglement in AutoVC's bottleneck embeddings. We demonstrate peculiar diversity between computational and human evaluations on singer-converted audio clips, which highlights the necessity of both. We also propose a pitch-matching mechanism between source and target singers to ensure these evaluations are not influenced by differences in pitch register.

Diffusion models are a class of deep generative models that have shown impressive results on various tasks with dense theoretical founding. Although diffusion models have achieved impressive quality and diversity of sample synthesis than other state-of-the-art models, they still suffer from costly sampling procedure and sub-optimal likelihood estimation. Recent studies have shown great enthusiasm on improving the performance of diffusion model. In this article, we present a first comprehensive review of existing variants of the diffusion models. Specifically, we provide a first taxonomy of diffusion models and categorize them variants to three types, namely sampling-acceleration enhancement, likelihood-maximization enhancement and data-generalization enhancement. We also introduce in detail other five generative models (i.e., variational autoencoders, generative adversarial networks, normalizing flow, autoregressive models, and energy-based models), and clarify the connections between diffusion models and these generative models. Then we make a thorough investigation into the applications of diffusion models, including computer vision, natural language processing, waveform signal processing, multi-modal modeling, molecular graph generation, time series modeling, and adversarial purification. Furthermore, we propose new perspectives pertaining to the development of this generative model.

Pre-trained deep neural network language models such as ELMo, GPT, BERT and XLNet have recently achieved state-of-the-art performance on a variety of language understanding tasks. However, their size makes them impractical for a number of scenarios, especially on mobile and edge devices. In particular, the input word embedding matrix accounts for a significant proportion of the model's memory footprint, due to the large input vocabulary and embedding dimensions. Knowledge distillation techniques have had success at compressing large neural network models, but they are ineffective at yielding student models with vocabularies different from the original teacher models. We introduce a novel knowledge distillation technique for training a student model with a significantly smaller vocabulary as well as lower embedding and hidden state dimensions. Specifically, we employ a dual-training mechanism that trains the teacher and student models simultaneously to obtain optimal word embeddings for the student vocabulary. We combine this approach with learning shared projection matrices that transfer layer-wise knowledge from the teacher model to the student model. Our method is able to compress the BERT_BASE model by more than 60x, with only a minor drop in downstream task metrics, resulting in a language model with a footprint of under 7MB. Experimental results also demonstrate higher compression efficiency and accuracy when compared with other state-of-the-art compression techniques.