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Diffusion Probabilistic Models stand as a critical tool in generative modelling, enabling the generation of complex data distributions. This family of generative models yields record-breaking performance in tasks such as image synthesis, video generation, and molecule design. Despite their capabilities, their efficiency, especially in the reverse process, remains a challenge due to slow convergence rates and high computational costs. In this paper, we introduce an approach that leverages continuous dynamical systems to design a novel denoising network for diffusion models that is more parameter-efficient, exhibits faster convergence, and demonstrates increased noise robustness. Experimenting with Denoising Diffusion Probabilistic Models (DDPMs), our framework operates with approximately a quarter of the parameters, and $\sim$ 30\% of the Floating Point Operations (FLOPs) compared to standard U-Nets in DDPMs. Furthermore, our model is notably faster in inference than the baseline when measured in fair and equal conditions. We also provide a mathematical intuition as to why our proposed reverse process is faster as well as a mathematical discussion of the empirical tradeoffs in the denoising downstream task. Finally, we argue that our method is compatible with existing performance enhancement techniques, enabling further improvements in efficiency, quality, and speed.

For a permutation $\pi: [K]\rightarrow [K]$, a sequence $f: \{1,2,\cdots, n\}\rightarrow \mathbb R$ contains a $\pi$-pattern of size $K$, if there is a sequence of indices $(i_1, i_2, \cdots, i_K)$ ($i_1<i_2<\cdots<i_K$), satisfying that $f(i_a)<f(i_b)$ if $\pi(a)<\pi(b)$, for $a,b\in [K]$. Otherwise, $f$ is referred to as $\pi$-free. For the special case where $\pi = (1,2,\cdots, K)$, it is referred to as the monotone pattern. \cite{newman2017testing} initiated the study of testing $\pi$-freeness with one-sided error. They focused on two specific problems, testing the monotone permutations and the $(1,3,2)$ permutation. For the problem of testing monotone permutation $(1,2,\cdots,K)$, \cite{ben2019finding} improved the $(\log n)^{O(K^2)}$ non-adaptive query complexity of \cite{newman2017testing} to $O((\log n)^{\lfloor \log_{2} K\rfloor})$. Further, \cite{ben2019optimal} proposed an adaptive algorithm with $O(\log n)$ query complexity. However, no progress has yet been made on the problem of testing $(1,3,2)$-freeness. In this work, we present an adaptive algorithm for testing $(1,3,2)$-freeness. The query complexity of our algorithm is $O(\epsilon^{-2}\log^4 n)$, which significantly improves over the $O(\epsilon^{-7}\log^{26}n)$-query adaptive algorithm of \cite{newman2017testing}. This improvement is mainly achieved by the proposal of a new structure embedded in the patterns.

Given a graph $G$, an integer $k\geq 0$, and a non-negative integral function $f:V(G) \rightarrow \mathcal{N}$, the {\sc Vector Domination} problem asks whether a set $S$ of vertices, of cardinality $k$ or less, exists in $G$ so that every vertex $v \in V(G)-S$ has at least $f(v)$ neighbors in $S$. The problem generalizes several domination problems and it has also been shown to generalize Bounded-Degree Vertex Deletion. In this paper, the parameterized version of Vector Domination is studied when the input graph is planar. A linear problem kernel is presented.

We provide an algorithm that maintains, against an adaptive adversary, a $(1-\varepsilon)$-approximate maximum matching in $n$-node $m$-edge general (not necessarily bipartite) undirected graph undergoing edge deletions with high probability with (amortized) $O(\mathrm{poly}(\varepsilon^{-1}, \log n))$ time per update. We also obtain the same update time for maintaining a fractional approximate weighted matching (and hence an approximation to the value of the maximum weight matching) and an integral approximate weighted matching in dense graphs. Our unweighted result improves upon the prior state-of-the-art which includes a $\mathrm{poly}(\log{n}) \cdot 2^{O(1/\varepsilon^2)}$ update time [Assadi-Bernstein-Dudeja 2022] and an $O(\sqrt{m} \varepsilon^{-2})$ update time [Gupta-Peng 2013], and our weighted result improves upon the $O(\sqrt{m}\varepsilon^{-O(1/\varepsilon)}\log{n})$ update time due to [Gupta-Peng 2013]. To obtain our results, we generalize a recent optimization approach to dynamic algorithms from [Jambulapati-Jin-Sidford-Tian 2022]. We show that repeatedly solving entropy-regularized optimization problems yields a lazy updating scheme for fractional decremental problems with a near-optimal number of updates. To apply this framework we develop optimization methods compatible with it and new dynamic rounding algorithms for the matching polytope.

In this paper, we consider the problem of maintaining a $(1-\varepsilon)$-approximate maximum weight matching in a dynamic graph $G$, while the adversary makes changes to the edges of the graph. In the fully dynamic setting, where both edge insertions and deletions are allowed, Gupta and Peng gave an algorithm for this problem with an update time of $\tilde{O}_{\varepsilon}(\sqrt{m})$. We study a natural relaxation of this problem, namely the decremental model, where the adversary is only allowed to delete edges. For the cardinality version of this problem in general (possibly, non-bipartite) graphs, Assadi, Bernstein, and Dudeja gave a decremental algorithm with update time $O_{\varepsilon}(\text{poly}(\log n))$. However, beating $\tilde{O}_{\varepsilon}(\sqrt{m})$ update time remained an open problem for the \emph{weighted} version in \emph{general graphs}. In this paper, we bridge the gap between unweighted and weighted general graphs for the decremental setting. We give a $O_{\varepsilon}(\text{poly}(\log n))$ update time algorithm that maintains a $(1-\varepsilon)$-approximate maximum weight matching under adversarial deletions. Like the decremental algorithm of Assadi, Bernstein, and Dudeja, our algorithm is randomized, but works against an adaptive adversary. It also matches the time bound for the cardinality version upto dependencies on $\varepsilon$ and a $\log R$ factor, where $R$ is the ratio between the maximum and minimum edge weight in $G$.

We give a stochastic optimization algorithm that solves a dense $n\times n$ real-valued linear system $Ax=b$, returning $\tilde x$ such that $\|A\tilde x-b\|\leq \epsilon\|b\|$ in time: $$\tilde O((n^2+nk^{\omega-1})\log1/\epsilon),$$ where $k$ is the number of singular values of $A$ larger than $O(1)$ times its smallest positive singular value, $\omega < 2.372$ is the matrix multiplication exponent, and $\tilde O$ hides a poly-logarithmic in $n$ factor. When $k=O(n^{1-\theta})$ (namely, $A$ has a flat-tailed spectrum, e.g., due to noisy data or regularization), this improves on both the cost of solving the system directly, as well as on the cost of preconditioning an iterative method such as conjugate gradient. In particular, our algorithm has an $\tilde O(n^2)$ runtime when $k=O(n^{0.729})$. We further adapt this result to sparse positive semidefinite matrices and least squares regression. Our main algorithm can be viewed as a randomized block coordinate descent method, where the key challenge is simultaneously ensuring good convergence and fast per-iteration time. In our analysis, we use theory of majorization for elementary symmetric polynomials to establish a sharp convergence guarantee when coordinate blocks are sampled using a determinantal point process. We then use a Markov chain coupling argument to show that similar convergence can be attained with a cheaper sampling scheme, and accelerate the block coordinate descent update via matrix sketching.

L1-norm regularized logistic regression models are widely used for analyzing data with binary response. In those analyses, fusing regression coefficients is useful for detecting groups of variables. This paper proposes a binomial logistic regression model with Bayesian fused lasso. Assuming a Laplace prior on regression coefficients and differences between adjacent regression coefficients enables us to perform variable selection and variable fusion simultaneously in the Bayesian framework. We also propose assuming a horseshoe prior on the differences to improve the flexibility of variable fusion. The Gibbs sampler is derived to estimate the parameters by a hierarchical expression of priors and a data-augmentation method. Using simulation studies and real data analysis, we compare the proposed methods with the existing method.

It is important to detect anomalous inputs when deploying machine learning systems. The use of larger and more complex inputs in deep learning magnifies the difficulty of distinguishing between anomalous and in-distribution examples. At the same time, diverse image and text data are available in enormous quantities. We propose leveraging these data to improve deep anomaly detection by training anomaly detectors against an auxiliary dataset of outliers, an approach we call Outlier Exposure (OE). This enables anomaly detectors to generalize and detect unseen anomalies. In extensive experiments on natural language processing and small- and large-scale vision tasks, we find that Outlier Exposure significantly improves detection performance. We also observe that cutting-edge generative models trained on CIFAR-10 may assign higher likelihoods to SVHN images than to CIFAR-10 images; we use OE to mitigate this issue. We also analyze the flexibility and robustness of Outlier Exposure, and identify characteristics of the auxiliary dataset that improve performance.

Recently, graph neural networks (GNNs) have revolutionized the field of graph representation learning through effectively learned node embeddings, and achieved state-of-the-art results in tasks such as node classification and link prediction. However, current GNN methods are inherently flat and do not learn hierarchical representations of graphs---a limitation that is especially problematic for the task of graph classification, where the goal is to predict the label associated with an entire graph. Here we propose DiffPool, a differentiable graph pooling module that can generate hierarchical representations of graphs and can be combined with various graph neural network architectures in an end-to-end fashion. DiffPool learns a differentiable soft cluster assignment for nodes at each layer of a deep GNN, mapping nodes to a set of clusters, which then form the coarsened input for the next GNN layer. Our experimental results show that combining existing GNN methods with DiffPool yields an average improvement of 5-10% accuracy on graph classification benchmarks, compared to all existing pooling approaches, achieving a new state-of-the-art on four out of five benchmark data sets.