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We present a survey of some of our recent results on Bayesian nonparametric inference for a multitude of stochastic processes. The common feature is that the prior distribution in the cases considered is on suitable sets of piecewise constant or piecewise linear functions, that differ for the specific situations at hand. Posterior consistency and in most cases contraction rates for the estimators are presented. Numerical studies on simulated and real data accompany the theoretical results.

In recent years, remarkable results have been achieved in self-supervised action recognition using skeleton sequences with contrastive learning. It has been observed that the semantic distinction of human action features is often represented by local body parts, such as legs or hands, which are advantageous for skeleton-based action recognition. This paper proposes an attention-based contrastive learning framework for skeleton representation learning, called SkeAttnCLR, which integrates local similarity and global features for skeleton-based action representations. To achieve this, a multi-head attention mask module is employed to learn the soft attention mask features from the skeletons, suppressing non-salient local features while accentuating local salient features, thereby bringing similar local features closer in the feature space. Additionally, ample contrastive pairs are generated by expanding contrastive pairs based on salient and non-salient features with global features, which guide the network to learn the semantic representations of the entire skeleton. Therefore, with the attention mask mechanism, SkeAttnCLR learns local features under different data augmentation views. The experiment results demonstrate that the inclusion of local feature similarity significantly enhances skeleton-based action representation. Our proposed SkeAttnCLR outperforms state-of-the-art methods on NTURGB+D, NTU120-RGB+D, and PKU-MMD datasets.

We consider the upper confidence bound strategy for Gaussian multi-armed bandits with known control horizon sizes $N$ and build its limiting description with a system of stochastic differential equations and ordinary differential equations. Rewards for the arms are assumed to have unknown expected values and known variances. A set of Monte-Carlo simulations was performed for the case of close distributions of rewards, when mean rewards differ by the magnitude of order $N^{-1/2}$, as it yields the highest normalized regret, to verify the validity of the obtained description. The minimal size of the control horizon when the normalized regret is not noticeably larger than maximum possible was estimated.

It is realized that existing powerful tests of goodness-of-fit are all based on sorted uniforms and, consequently, can suffer from the confounded effect of different locations and various signal frequencies in the deviations of the distributions under the alternative hypothesis from those under the null. This paper proposes circularly symmetric tests that are obtained by circularizing reweighted Anderson-Darling tests, with the focus on the circularized versions of Anderson-Darling and Zhang test statistics. Two specific types of circularization are considered, one is obtained by taking the average of the corresponding so-called scan test statistics and the other by using the maximum. To a certain extent, this circularization technique effectively eliminates the location effect and allows the weights to focus on the various signal frequencies. A limited but arguably convincing simulation study on finite-sample performance demonstrates that the circularized Zhang method outperforms the circularized Anderson-Darling and that the circularized tests outperform their parent methods. Large-sample theoretical results are also obtained for the average type of circularization. The results show that both the circularized Anderson-Darling and circularized Zhang have asymptotic distributions that are a weighted sum of an infinite number of independent squared standard normal random variables. In addition, the kernel matrices and functions are circulant. As a result, asymptotic approximations are computationally efficient via the fast Fourier transform.

Let $G$ be a graph on $n$ vertices of maximum degree $\Delta$. We show that, for any $\delta > 0$, the down-up walk on independent sets of size $k \leq (1-\delta)\alpha_c(\Delta)n$ mixes in time $O_{\Delta,\delta}(k\log{n})$, thereby resolving a conjecture of Davies and Perkins in an optimal form. Here, $\alpha_{c}(\Delta)n$ is the NP-hardness threshold for the problem of counting independent sets of a given size in a graph on $n$ vertices of maximum degree $\Delta$. Our mixing time has optimal dependence on $k,n$ for the entire range of $k$; previously, even polynomial mixing was not known. In fact, for $k = \Omega_{\Delta}(n)$ in this range, we establish a log-Sobolev inequality with optimal constant $\Omega_{\Delta,\delta}(1/n)$. At the heart of our proof are three new ingredients, which may be of independent interest. The first is a method for lifting $\ell_\infty$-independence from a suitable distribution on the discrete cube -- in this case, the hard-core model -- to the slice by proving stability of an Edgeworth expansion using a multivariate zero-free region for the base distribution. The second is a generalization of the Lee-Yau induction to prove log-Sobolev inequalities for distributions on the slice with considerably less symmetry than the uniform distribution. The third is a sharp decomposition-type result which provides a lossless comparison between the Dirichlet form of the original Markov chain and that of the so-called projected chain in the presence of a contractive coupling.

Given $\mathbf A \in \mathbb{R}^{n \times n}$ with entries bounded in magnitude by $1$, it is well-known that if $S \subset [n] \times [n]$ is a uniformly random subset of $\tilde{O} (n/\epsilon^2)$ entries, and if ${\mathbf A}_S$ equals $\mathbf A$ on the entries in $S$ and is zero elsewhere, then $\|\mathbf A - \frac{n^2}{s} \cdot {\mathbf A}_S\|_2 \le \epsilon n$ with high probability, where $\|\cdot\|_2$ is the spectral norm. We show that for positive semidefinite (PSD) matrices, no randomness is needed at all in this statement. Namely, there exists a fixed subset $S$ of $\tilde{O} (n/\epsilon^2)$ entries that acts as a universal sparsifier: the above error bound holds simultaneously for every bounded entry PSD matrix $\mathbf A \in \mathbb{R}^{n \times n}$. One can view this result as a significant extension of a Ramanujan expander graph, which sparsifies any bounded entry PSD matrix, not just the all ones matrix. We leverage the existence of such universal sparsifiers to give the first deterministic algorithms for several central problems related to singular value computation that run in faster than matrix multiplication time. We also prove universal sparsification bounds for non-PSD matrices, showing that $\tilde{O} (n/\epsilon^4)$ entries suffices to achieve error $\epsilon \cdot \max(n,\|\mathbf A\|_1)$, where $\|\mathbf A\|_1$ is the trace norm. We prove that this is optimal up to an $\tilde{O} (1/\epsilon^2)$ factor. Finally, we give an improved deterministic spectral approximation algorithm for PSD $\mathbf A$ with entries lying in $\{-1,0,1\}$, which we show is nearly information-theoretically optimal.

What matters for contrastive learning? We argue that contrastive learning heavily relies on informative features, or "hard" (positive or negative) features. Early works include more informative features by applying complex data augmentations and large batch size or memory bank, and recent works design elaborate sampling approaches to explore informative features. The key challenge toward exploring such features is that the source multi-view data is generated by applying random data augmentations, making it infeasible to always add useful information in the augmented data. Consequently, the informativeness of features learned from such augmented data is limited. In response, we propose to directly augment the features in latent space, thereby learning discriminative representations without a large amount of input data. We perform a meta learning technique to build the augmentation generator that updates its network parameters by considering the performance of the encoder. However, insufficient input data may lead the encoder to learn collapsed features and therefore malfunction the augmentation generator. A new margin-injected regularization is further added in the objective function to avoid the encoder learning a degenerate mapping. To contrast all features in one gradient back-propagation step, we adopt the proposed optimization-driven unified contrastive loss instead of the conventional contrastive loss. Empirically, our method achieves state-of-the-art results on several benchmark datasets.

In this paper, we propose a novel Feature Decomposition and Reconstruction Learning (FDRL) method for effective facial expression recognition. We view the expression information as the combination of the shared information (expression similarities) across different expressions and the unique information (expression-specific variations) for each expression. More specifically, FDRL mainly consists of two crucial networks: a Feature Decomposition Network (FDN) and a Feature Reconstruction Network (FRN). In particular, FDN first decomposes the basic features extracted from a backbone network into a set of facial action-aware latent features to model expression similarities. Then, FRN captures the intra-feature and inter-feature relationships for latent features to characterize expression-specific variations, and reconstructs the expression feature. To this end, two modules including an intra-feature relation modeling module and an inter-feature relation modeling module are developed in FRN. Experimental results on both the in-the-lab databases (including CK+, MMI, and Oulu-CASIA) and the in-the-wild databases (including RAF-DB and SFEW) show that the proposed FDRL method consistently achieves higher recognition accuracy than several state-of-the-art methods. This clearly highlights the benefit of feature decomposition and reconstruction for classifying expressions.

Graph Neural Networks (GNNs) have been shown to be effective models for different predictive tasks on graph-structured data. Recent work on their expressive power has focused on isomorphism tasks and countable feature spaces. We extend this theoretical framework to include continuous features - which occur regularly in real-world input domains and within the hidden layers of GNNs - and we demonstrate the requirement for multiple aggregation functions in this context. Accordingly, we propose Principal Neighbourhood Aggregation (PNA), a novel architecture combining multiple aggregators with degree-scalers (which generalize the sum aggregator). Finally, we compare the capacity of different models to capture and exploit the graph structure via a novel benchmark containing multiple tasks taken from classical graph theory, alongside existing benchmarks from real-world domains, all of which demonstrate the strength of our model. With this work, we hope to steer some of the GNN research towards new aggregation methods which we believe are essential in the search for powerful and robust models.

Graph convolutional networks (GCNs) have been successfully applied in node classification tasks of network mining. However, most of these models based on neighborhood aggregation are usually shallow and lack the "graph pooling" mechanism, which prevents the model from obtaining adequate global information. In order to increase the receptive field, we propose a novel deep Hierarchical Graph Convolutional Network (H-GCN) for semi-supervised node classification. H-GCN first repeatedly aggregates structurally similar nodes to hyper-nodes and then refines the coarsened graph to the original to restore the representation for each node. Instead of merely aggregating one- or two-hop neighborhood information, the proposed coarsening procedure enlarges the receptive field for each node, hence more global information can be learned. Comprehensive experiments conducted on public datasets demonstrate the effectiveness of the proposed method over the state-of-art methods. Notably, our model gains substantial improvements when only a few labeled samples are provided.