We introduce a tensor-based clustering method to extract sparse, low-dimensional structure from high-dimensional, multi-indexed datasets. Specifically, this framework is designed to enable detection of clusters of data in the presence of structural requirements which we encode as algebraic constraints in a linear program. We illustrate our method on a collection of experiments measuring the response of genetically diverse breast cancer cell lines to an array of ligands. Each experiment consists of a cell line-ligand combination, and contains time-course measurements of the early-signalling kinases MAPK and AKT at two different ligand dose levels. By imposing appropriate structural constraints and respecting the multi-indexed structure of the data, our clustering analysis can be optimized for biological interpretation and therapeutic understanding. We then perform a systematic, large-scale exploration of mechanistic models of MAPK-AKT crosstalk for each cluster. This analysis allows us to quantify the heterogeneity of breast cancer cell subtypes, and leads to hypotheses about the mechanisms by which cell lines respond to ligands. Our clustering method is general and can be tailored to a variety of applications in science and industry.

Good profitable customers rarely become unprofitable. It is more likely that they were unprofitable from the onset. Determining an approach to define customer value can be a complex decision. Traditionally, we use gross margin in identifying good and bad customers. For example, if your overhead costs are 25% of gross revenue, a good customer is anyone with a gross margin over 25%.

In recent years there is a growing interest in using deep representations for reinforcement learning. In this paper, we present a methodology and tools to analyze Deep Q-networks (DQNs) in a non-blind matter. Moreover, we propose a new model, the Semi Aggregated Markov Decision Process (SAMDP), and an algorithm that learns it automatically. The SAMDP model allows us to identify spatio-temporal abstractions directly from features and may be used as a sub-goal detector in future work. Using our tools we reveal that the features learned by DQNs aggregate the state space in a hierarchical fashion, explaining its success. Moreover, we are able to understand and describe the policies learned by DQNs for three different Atari2600 games and suggest ways to interpret, debug and optimize deep neural networks in reinforcement learning.

While there's not necessarily a "correct" answer here, it's most likely you split the bugs into four clusters. That wasn't too bad, was it? You could probably do the same with twice as many bugs, right? If you had a bit of time to spare -- or a passion for entomology -- you could probably even do the same with a hundred bugs. For a machine though, grouping ten objects into however many meaningful clusters is no small task, thanks to a mind-bending branch of maths called combinatorics, which tells us that are 115,975 different possible ways you could have grouped those ten insects together. Had there been twenty bugs, there would have been over fifty trillion possible ways of clustering them. With a hundred bugs -- there'd be many times more solutions than there are particles in the known universe. In fact, there are more than four million billion googol solutions (what's a googol?).

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When faced with high frequency streams of data, clustering raises theoretical and algorithmic pitfalls. We introduce a new and adaptive online clustering algorithm relying on a quasi-Bayesian approach, with a dynamic (\emph{i.e.}, time-dependent) estimation of the (unknown and changing) number of clusters. We prove that our approach is supported by minimax regret bounds. We also provide an RJMCMC-flavored implementation (called PACBO) for which we give a convergence guarantee. Finally, numerical experiments illustrate the potential of our procedure.

In this paper, we introduce an algorithm for performing spectral clustering efficiently. Spectral clustering is a powerful clustering algorithm that suffers from high computational complexity, due to eigen decomposition. In this work, we first build the adjacency matrix of the corresponding graph of the dataset. To build this matrix, we only consider a limited number of points, called landmarks, and compute the similarity of all data points with the landmarks. Then, we present a definition of the Laplacian matrix of the graph that enable us to perform eigen decomposition efficiently, using a deep autoencoder. The overall complexity of the algorithm for eigen decomposition is $O(np)$, where $n$ is the number of data points and $p$ is the number of landmarks. At last, we evaluate the performance of the algorithm in different experiments.

Motivation: Single cell transcriptome sequencing (scRNA-Seq) has become a revolutionary tool to study cellular and molecular processes at single cell resolution. Among existing technologies, the recently developed droplet-based platform enables efficient parallel processing of thousands of single cells with direct counting of transcript copies using Unique Molecular Identifier (UMI). Despite the technology advances, statistical methods and computational tools are still lacking for analyzing droplet-based scRNA-Seq data. Particularly, model-based approaches for clustering large-scale single cell transcriptomic data are still under-explored. Methods: We developed DIMM-SC, a Dirichlet Mixture Model for clustering droplet-based Single Cell transcriptomic data. This approach explicitly models UMI count data from scRNA-Seq experiments and characterizes variations across different cell clusters via a Dirichlet mixture prior. An expectation-maximization algorithm is used for parameter inference. Results: We performed comprehensive simulations to evaluate DIMM-SC and compared it with existing clustering methods such as K-means, CellTree and Seurat. In addition, we analyzed public scRNA-Seq datasets with known cluster labels and in-house scRNA-Seq datasets from a study of systemic sclerosis with prior biological knowledge to benchmark and validate DIMM-SC. Both simulation studies and real data applications demonstrated that overall, DIMM-SC achieves substantially improved clustering accuracy and much lower clustering variability compared to other existing clustering methods. More importantly, as a model-based approach, DIMM-SC is able to quantify the clustering uncertainty for each single cell, facilitating rigorous statistical inference and biological interpretations, which are typically unavailable from existing clustering methods.

Cluster analysis of very high dimensional data can benefit from the properties of such high dimensionality. Informally expressed, in this work, our focus is on the analogous situation when the dimensionality is moderate to small, relative to a massively sized set of observations. Mathematically expressed, these are the dual spaces of observations and attributes. The point cloud of observations is in attribute space, and the point cloud of attributes is in observation space. In this paper, we begin by summarizing various perspectives related to methodologies that are used in multivariate analytics. We draw on these to establish an efficient clustering processing pipeline, both partitioning and hierarchical clustering.

Many modern clustering methods scale well to a large number of data items, N, but not to a large number of clusters, K. This paper introduces PERCH, a new non-greedy algorithm for online hierarchical clustering that scales to both massive N and K--a problem setting we term extreme clustering. Our algorithm efficiently routes new data points to the leaves of an incrementally-built tree. Motivated by the desire for both accuracy and speed, our approach performs tree rotations for the sake of enhancing subtree purity and encouraging balancedness. We prove that, under a natural separability assumption, our non-greedy algorithm will produce trees with perfect dendrogram purity regardless of online data arrival order. Our experiments demonstrate that PERCH constructs more accurate trees than other tree-building clustering algorithms and scales well with both N and K, achieving a higher quality clustering than the strongest flat clustering competitor in nearly half the time.