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Sum-of-norms clustering is a convex optimization problem whose solution can be used for the clustering of multivariate data. We propose and study a localized version of this method, and show in particular that it can separate arbitrarily close balls in the stochastic ball model. More precisely, we prove a quantitative bound on the error incurred in the clustering of disjoint connected sets. Our bound is expressed in terms of the number of datapoints and the localization length of the functional.

In a previous paper we have presented a CEGAR approach for the verification of parameterized systems with an arbitrary number of processes organized in an array or a ring. The technique is based on the iterative computation of parameterized invariants, i.e., infinite families of invariants for the infinitely many instances of the system. Safety properties are proved by checking that every global configuration of the system satisfying all parameterized invariants also satisfies the property; we have shown that this check can be reduced to the satisfiability problem for Monadic Second Order on words, which is decidable. A strong limitation of the approach is that processes can only have a fixed number of variables with a fixed finite range. In particular, they cannot use variables with range [0,N-1], where N is the number of processes, which appear in many standard distributed algorithms. In this paper, we extend our technique to this case. While conducting the check whether a safety property is inductive assuming a computed set of invariants becomes undecidable, we show how to reduce it to checking satisfiability of a first-order formula. We report on experiments showing that automatic first-order theorem provers can still perform this check for a collection of non-trivial examples. Additionally, we can give small sets of readable invariants for these checks.

We present two lower bounds on sub-packetization level $\alpha$ of MSR codes with parameters $(n, k, d=n-1, \alpha)$ where $n$ is the block length, $k$ dimension, $d$ number of helper nodes contacted during single node repair and $\alpha$ the sub-packetization level. The first bound we present is for any MSR code and is given by $\alpha \ge e^{\frac{(k-1)(r-1)}{2r^2}}$. The second bound we present is for the case of optimal-access MSR codes and the bound is given by $\alpha \ge \min \{ r^{\frac{n-1}{r}}, r^{k-1} \}$. There exist optimal-access MSR constructions that achieve the second sub-packetization level bound with an equality making this bound tight. We also prove that for an optimal-access MSR codes to have optimal sub-packetization level under the constraint that the indices of helper symbols are dependant only on the failed node, it is needed that the support of the parity check matrix is same as the support structure of several other optimal constructions in literature.

Within a Bayesian framework, a comprehensive investigation of mixtures of finite mixtures (MFMs), i.e., finite mixtures with a prior on the number of components, is performed. This model class has applications in model-based clustering as well as for semi-parametric density estimation and requires suitable prior specifications and inference methods to exploit its full potential. We contribute by considering a generalized class of MFMs where the hyperparameter $\gamma_K$ of a symmetric Dirichlet prior on the weight distribution depends on the number of components. We show that this model class may be regarded as a Bayesian non-parametric mixture outside the class of Gibbs-type priors. We emphasize the distinction between the number of components $K$ of a mixture and the number of clusters $K_+$, i.e., the number of filled components given the data. In the MFM model, $K_+$ is a random variable and its prior depends on the prior on $K$ and on the hyperparameter $\gamma_K$. We employ a flexible prior distribution for the number of components $K$ and derive the corresponding prior on the number of clusters $K_+$ for generalized MFMs. For posterior inference, we propose the novel telescoping sampler which allows Bayesian inference for mixtures with arbitrary component distributions without resorting to reversible jump Markov chain Monte Carlo (MCMC) methods. The telescoping sampler explicitly samples the number of components, but otherwise requires only the usual MCMC steps of a finite mixture model. The ease of its application using different component distributions is demonstrated on several data sets.

We consider a minimal residual discretization of a simultaneous space-time variational formulation of parabolic evolution equations. Under the usual `LBB' stability condition on pairs of trial- and test spaces we show quasi-optimality of the numerical approximations without assuming symmetry of the spatial part of the differential operator. Under a stronger LBB condition we show error estimates in an energy-norm which are independent of this spatial differential operator.

This paper studies task adaptive pre-trained model selection, an \emph{underexplored} problem of assessing pre-trained models so that models suitable for the task can be selected from the model zoo without fine-tuning. A pilot work~\cite{nguyen_leep:_2020} addressed the problem in transferring supervised pre-trained models to classification tasks, but it cannot handle emerging unsupervised pre-trained models or regression tasks. In pursuit of a practical assessment method, we propose to estimate the maximum evidence (marginalized likelihood) of labels given features extracted by pre-trained models. The maximum evidence is \emph{less prone to over-fitting} than the likelihood, and its \emph{expensive computation can be dramatically reduced} by our carefully designed algorithm. The Logarithm of Maximum Evidence (LogME) can be used to assess pre-trained models for transfer learning: a pre-trained model with high LogME is likely to have good transfer performance. LogME is fast, accurate, and general, characterizing it as \emph{the first practical assessment method for transfer learning}. Compared to brute-force fine-tuning, LogME brings over $3000\times$ speedup in wall-clock time. It outperforms prior methods by a large margin in their setting and is applicable to new settings that prior methods cannot deal with. It is general enough to diverse pre-trained models (supervised pre-trained and unsupervised pre-trained), downstream tasks (classification and regression), and modalities (vision and language). Code is at \url{//github.com/thuml/LogME}.

Latent Dirichlet Allocation (LDA) is a topic model widely used in natural language processing and machine learning. Most approaches to training the model rely on iterative algorithms, which makes it difficult to run LDA on big corpora that are best analyzed in parallel and distributed computational environments. Indeed, current approaches to parallel inference either don't converge to the correct posterior or require storage of large dense matrices in memory. We present a novel sampler that overcomes both problems, and we show that this sampler is faster, both empirically and theoretically, than previous Gibbs samplers for LDA. We do so by employing a novel P\'olya-urn-based approximation in the sparse partially collapsed sampler for LDA. We prove that the approximation error vanishes with data size, making our algorithm asymptotically exact, a property of importance for large-scale topic models. In addition, we show, via an explicit example, that -- contrary to popular belief in the topic modeling literature -- partially collapsed samplers can be more efficient than fully collapsed samplers. We conclude by comparing the performance of our algorithm with that of other approaches on well-known corpora.

The Pachinko Allocation Machine (PAM) is a deep topic model that allows representing rich correlation structures among topics by a directed acyclic graph over topics. Because of the flexibility of the model, however, approximate inference is very difficult. Perhaps for this reason, only a small number of potential PAM architectures have been explored in the literature. In this paper we present an efficient and flexible amortized variational inference method for PAM, using a deep inference network to parameterize the approximate posterior distribution in a manner similar to the variational autoencoder. Our inference method produces more coherent topics than state-of-art inference methods for PAM while being an order of magnitude faster, which allows exploration of a wider range of PAM architectures than have previously been studied.

We consider the task of learning the parameters of a {\em single} component of a mixture model, for the case when we are given {\em side information} about that component, we call this the "search problem" in mixture models. We would like to solve this with computational and sample complexity lower than solving the overall original problem, where one learns parameters of all components. Our main contributions are the development of a simple but general model for the notion of side information, and a corresponding simple matrix-based algorithm for solving the search problem in this general setting. We then specialize this model and algorithm to four common scenarios: Gaussian mixture models, LDA topic models, subspace clustering, and mixed linear regression. For each one of these we show that if (and only if) the side information is informative, we obtain parameter estimates with greater accuracy, and also improved computation complexity than existing moment based mixture model algorithms (e.g. tensor methods). We also illustrate several natural ways one can obtain such side information, for specific problem instances. Our experiments on real data sets (NY Times, Yelp, BSDS500) further demonstrate the practicality of our algorithms showing significant improvement in runtime and accuracy.

Discrete random structures are important tools in Bayesian nonparametrics and the resulting models have proven effective in density estimation, clustering, topic modeling and prediction, among others. In this paper, we consider nested processes and study the dependence structures they induce. Dependence ranges between homogeneity, corresponding to full exchangeability, and maximum heterogeneity, corresponding to (unconditional) independence across samples. The popular nested Dirichlet process is shown to degenerate to the fully exchangeable case when there are ties across samples at the observed or latent level. To overcome this drawback, inherent to nesting general discrete random measures, we introduce a novel class of latent nested processes. These are obtained by adding common and group-specific completely random measures and, then, normalising to yield dependent random probability measures. We provide results on the partition distributions induced by latent nested processes, and develop an Markov Chain Monte Carlo sampler for Bayesian inferences. A test for distributional homogeneity across groups is obtained as a by product. The results and their inferential implications are showcased on synthetic and real data.