$k$-means clustering is a fundamental problem in unsupervised learning. The problem concerns finding a partition of the data points into $k$ clusters such that the within-cluster variation is minimized. Despite its importance and wide applicability, a theoretical understanding of the $k$-means problem has not been completely satisfactory. Existing algorithms with theoretical performance guarantees often rely on sophisticated (sometimes artificial) algorithmic techniques and restricted assumptions on the data. The main challenge lies in the non-convex nature of the problem; in particular, there exist additional local solutions other than the global optimum. Moreover, the simplest and most popular algorithm for $k$-means, namely Lloyd's algorithm, generally converges to such spurious local solutions both in theory and in practice. In this paper, we approach the $k$-means problem from a new perspective, by investigating the structures of these spurious local solutions under a probabilistic generative model with $k$ ground truth clusters. As soon as $k=3$, spurious local minima provably exist, even for well-separated and balanced clusters. One such local minimum puts two centers at one true cluster, and the third center in the middle of the other two true clusters. For general $k$, one local minimum puts multiple centers at a true cluster, and one center in the middle of multiple true clusters. Perhaps surprisingly, we prove that this is essentially the only type of spurious local minima under a separation condition. Our results pertain to the $k$-means formulation for mixtures of Gaussians or bounded distributions. Our theoretical results corroborate existing empirical observations and provide justification for several improved algorithms for $k$-means clustering.

One promising approach to dealing with datapoints that are outside of the initial training distribution (OOD) is to create new classes that capture similarities in the datapoints previously rejected as uncategorizable. Systems that generate labels can be deployed against an arbitrary amount of data, discovering classification schemes that through training create a higher quality representation of data. We introduce the Dataset Reconstruction Accuracy, a new and important measure of the effectiveness of a model's ability to create labels. We introduce benchmarks against this Dataset Reconstruction metric. We apply a new heuristic, class learnability, for deciding whether a class is worthy of addition to the training dataset. We show that our class discovery system can be successfully applied to vision and language, and we demonstrate the value of semi-supervised learning in automatically discovering novel classes.

Sparsity in Deep Neural Networks (DNNs) is studied extensively with the focus of maximizing prediction accuracy given an overall parameter budget. Existing methods rely on uniform or heuristic non-uniform sparsity budgets which have sub-optimal layer-wise parameter allocation resulting in a) lower prediction accuracy or b) higher inference cost (FLOPs). This work proposes Soft Threshold Reparameterization (STR), a novel use of the soft-threshold operator on DNN weights. STR smoothly induces sparsity while learning pruning thresholds thereby obtaining a non-uniform sparsity budget. Our method achieves state-of-the-art accuracy for unstructured sparsity in CNNs (ResNet50 and MobileNetV1 on ImageNet-1K), and, additionally, learns non-uniform budgets that empirically reduce the FLOPs by up to 50%. Notably, STR boosts the accuracy over existing results by up to 10% in the ultra sparse (99%) regime and can also be used to induce low-rank (structured sparsity) in RNNs. In short, STR is a simple mechanism which learns effective sparsity budgets that contrast with popular heuristics.

Big Data is a massive volume of both structured and unstructured data that is too large and it also difficult to process using traditional techniques. Clustering algorithms have developed as a powerful learning tool that can exactly analyze the volume of data that produced by modern applications. Clustering in data mining is the grouping of a particular set of objects based on their characteristics. The main aim of clustering is to classified data into clusters such that objects are grouped in the same clusters when they are corresponding according to similarities and features mainly. Till now, K-MEANS is the best utilized calculation connected in a wide scope of zones to recognize gatherings where cluster separations are a lot than between gathering separations. Our developed algorithm works with K-MEANS for high quality clustering during clustering from big data. Our proposed algorithm EG K-MEANS : Extended Generation K-MEANS solves mainly three issues of K-MEANS: unhealthy initialization, dynamic centroid selection and empty clustering. It ensures the best way of preventing unhealthy initialization, dynamic centroid selection and empty clustering problems for getting high quality clustering.

In recent years, a great variety of nature- and bio-inspired algorithms has been reported in the literature. This algorithmic family simulates different biological processes observed in Nature in order to efficiently address complex optimization problems. In the last years the number of bio-inspired optimization approaches in literature has grown considerably, reaching unprecedented levels that dark the future prospects of this field of research. This paper addresses this problem by proposing two comprehensive, principle-based taxonomies that allow researchers to organize existing and future algorithmic developments into well-defined categories, considering two different criteria: the source of inspiration and the behavior of each algorithm. Using these taxonomies we review more than three hundred publications dealing with nature-inspired and bio-inspired algorithms, and proposals falling within each of these categories are examined, leading to a critical summary of design trends and similarities between them, and the identification of the most similar classical algorithm for each reviewed paper. From our analysis we conclude that a poor relationship is often found between the natural inspiration of an algorithm and its behavior. Furthermore, similarities in terms of behavior between different algorithms are greater than what is claimed in their public disclosure: specifically, we show that more than one-third of the reviewed bio-inspired solvers are versions of classical algorithms. Grounded on the conclusions of our critical analysis, we give several recommendations and points of improvement for better methodological practices in this active and growing research field.

Representing graphs as sets of node embeddings in certain curved Riemannian manifolds has recently gained momentum in machine learning due to their desirable geometric inductive biases, e.g., hierarchical structures benefit from hyperbolic geometry. However, going beyond embedding spaces of constant sectional curvature, while potentially more representationally powerful, proves to be challenging as one can easily lose the appeal of computationally tractable tools such as geodesic distances or Riemannian gradients. Here, we explore computationally efficient matrix manifolds, showcasing how to learn and optimize graph embeddings in these Riemannian spaces. Empirically, we demonstrate consistent improvements over Euclidean geometry while often outperforming hyperbolic and elliptical embeddings based on various metrics that capture different graph properties. Our results serve as new evidence for the benefits of non-Euclidean embeddings in machine learning pipelines.

We propose a new Stein self-repulsive dynamics for obtaining diversified samples from intractable un-normalized distributions. Our idea is to introduce Stein variational gradient as a repulsive force to push the samples of Langevin dynamics away from the past trajectories. This simple idea allows us to significantly decrease the auto-correlation in Langevin dynamics and hence increase the effective sample size. Importantly, as we establish in our theoretical analysis, the asymptotic stationary distribution remains correct even with the addition of the repulsive force, thanks to the special properties of the Stein variational gradient. We perform extensive empirical studies of our new algorithm, showing that our method yields much higher sample efficiency and better uncertainty estimation than vanilla Langevin dynamics.

We consider the problem of online prediction in a marginally stable linear dynamical system subject to bounded adversarial or (non-isotropic) stochastic perturbations. This poses two challenges. Firstly, the system is in general unidentifiable, so recent and classical results on parameter recovery do not apply. Secondly, because we allow the system to be marginally stable, the state can grow polynomially with time; this causes standard regret bounds in online convex optimization to be vacuous. In spite of these challenges, we show that the online least-squares algorithm achieves sublinear regret (improvable to polylogarithmic in the stochastic setting), with polynomial dependence on the system's parameters. This requires a refined regret analysis, including a structural lemma showing the current state of the system to be a small linear combination of past states, even if the state grows polynomially. By applying our techniques to learning an autoregressive filter, we also achieve logarithmic regret in the partially observed setting under Gaussian noise, with polynomial dependence on the memory of the associated Kalman filter.

This work examines the convergence of stochastic gradient-based optimization algorithms that use early stopping based on a validation function. The form of early stopping we consider is that optimization terminates when the norm of the gradient of a validation function falls below a threshold. We derive conditions that guarantee this stopping rule is well-defined, and provide bounds on the expected number of iterations and gradient evaluations needed to meet this criterion. The guarantee accounts for the distance between the training and validation sets, measured with the Wasserstein distance. We develop the approach in the general setting of a first-order optimization algorithm, with possibly biased update directions subject to a geometric drift condition. We then derive bounds on the expected running time for early stopping variants of several algorithms, including stochastic gradient descent (SGD), decentralized SGD (DSGD), and the stochastic variance reduced gradient (SVRG) algorithm. Finally, we consider the generalization properties of the iterate returned by early stopping.

Aspect based Sentiment Analysis is a major subarea of sentiment analysis. Many supervised and unsupervised approaches have been proposed in the past for detecting and analyzing the sentiment of aspect terms. In this paper, a graph-based semi-supervised learning approach for aspect term extraction is proposed. In this approach, every identified token in the review document is classified as aspect or non-aspect term from a small set of labeled tokens using label spreading algorithm. The k-Nearest Neighbor (kNN) for graph sparsification is employed in the proposed approach to make it more time and memory efficient. The proposed work is further extended to determine the polarity of the opinion words associated with the identified aspect terms in review sentence to generate visual aspect-based summary of review documents. The experimental study is conducted on benchmark and crawled datasets of restaurant and laptop domains with varying value of labeled instances. The results depict that the proposed approach could achieve good result in terms of Precision, Recall and Accuracy with limited availability of labeled data.