Explainable clustering by axis-aligned decision trees was introduced by Moshkovitz et al. (2020) and has gained considerable interest. Prior work has focused on minimizing the price of explainability for specific clustering objectives, lacking a general method to fit an explanation tree to any given clustering, without restrictions. In this work, we propose a new and generic approach to explainable clustering, based on spectral graph partitioning. With it, we design an explainable clustering algorithm that can fit an explanation tree to any given non-explainable clustering, or directly to the dataset itself. Moreover, we show that prior algorithms can also be interpreted as graph partitioning, through a generalized framework due to Trevisan (2013) wherein cuts are optimized in two graphs simultaneously. Our experiments show the favorable performance of our method compared to baselines on a range of datasets.

Do the main claims made in the abstract and introduction accurately reflect the paper's Did you discuss any potential negative societal impacts of your work? Such methods could provide biased representations that could have negative downstream use cases as features for models, in search (for example). Did you include complete proofs of all theoretical results? Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Y es] (b) Did you specify all the training details (e.g., data splits, hyperparameters, how they Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? Did you include the total amount of compute and the type of resources used (e.g., type Did you include any new assets either in the supplemental material or as a URL? [Y es] Did you discuss whether and how consent was obtained from people whose data you're using/curating?

We revisit the well-studied problem of approximating a matrix product, $\mathbf{A}^T\mathbf{B}$, based on small space sketches $\mathcal{S}(\mathbf{A})$ and $\mathcal{S}(\mathbf{B})$ of $\mathbf{A} \in \R^{n \times d}$ and $\mathbf{B}\in \R^{n \times m}$. We are interested in the setting where the sketches must be computed independently of each other, except for the use of a shared random seed. We prove that, when $\mathbf{A}$ and $\mathbf{B}$ are sparse, methods based on \emph{coordinated random sampling} can outperform classical linear sketching approaches, like Johnson-Lindenstrauss Projection or CountSketch. For example, to obtain Frobenius norm error $\epsilon\|\mathbf{A}\|_F\|\mathbf{B}\|_F$, coordinated sampling requires sketches of size $O(s/\epsilon^2)$ when $\mathbf{A}$ and $\mathbf{B}$ have at most $s \leq d,m$ non-zeros per row. In contrast, linear sketching leads to sketches of size $O(d/\epsilon^2)$ and $O(m/\epsilon^2)$ for $\mathbf{A}$ and $\mathbf{B}$. We empirically evaluate our approach on two applications: 1) distributed linear regression in databases, a problem motivated by tasks like dataset discovery and augmentation, and 2) approximating attention matrices in transformer-based language models. In both cases, our sampling algorithms yield an order of magnitude improvement over linear sketching.

This book introduces the mathematical foundations and techniques that lead to the development and analysis of many of the algorithms that are used in machine learning. It starts with an introductory chapter that describes notation used throughout the book and serve at a reminder of basic concepts in calculus, linear algebra and probability and also introduces some measure theoretic terminology, which can be used as a reading guide for the sections that use these tools. The introductory chapters also provide background material on matrix analysis and optimization. The latter chapter provides theoretical support to many algorithms that are used in the book, including stochastic gradient descent, proximal methods, etc. After discussing basic concepts for statistical prediction, the book includes an introduction to reproducing kernel theory and Hilbert space techniques, which are used in many places, before addressing the description of various algorithms for supervised statistical learning, including linear methods, support vector machines, decision trees, boosting, or neural networks. The subject then switches to generative methods, starting with a chapter that presents sampling methods and an introduction to the theory of Markov chains. The following chapter describe the theory of graphical models, an introduction to variational methods for models with latent variables, and to deep-learning based generative models. The next chapters focus on unsupervised learning methods, for clustering, factor analysis and manifold learning. The final chapter of the book is theory-oriented and discusses concentration inequalities and generalization bounds.

We consider the problem of finding good low rank approximations of symmetric, positive-definite $A \in \mathbb{R}^{n \times n}$. Chen-Epperly-Tropp-Webber showed, among many other things, that the randomly pivoted partial Cholesky algorithm that chooses the $i-$th row with probability proportional to the diagonal entry $A_{ii}$ leads to a universal contraction of the trace norm (the Schatten 1-norm) in expectation for each step. We show that if one chooses the $i-$th row with likelihood proportional to $A_{ii}^2$ one obtains the same result in the Frobenius norm (the Schatten 2-norm). Implications for the greedy pivoting rule and pivot selection strategies are discussed.

Entity resolution (record linkage, microclustering) systems are notoriously difficult to evaluate. Looking for a needle in a haystack, traditional evaluation methods use sophisticated, application-specific sampling schemes to find matching pairs of records among an immense number of non-matches. We propose an alternative that facilitates the creation of representative, reusable benchmark data sets without necessitating complex sampling schemes. These benchmark data sets can then be used for model training and a variety of evaluation tasks. Specifically, we propose an entity-centric data labeling methodology that integrates with a unified framework for monitoring summary statistics, estimating key performance metrics such as cluster and pairwise precision and recall, and analyzing root causes for errors. We validate the framework in an application to inventor name disambiguation and through simulation studies. Software: https://github.com/OlivierBinette/er-evaluation/

This paper introduces a novel evaluation methodology for entity resolution algorithms. It is motivated by PatentsView.org, a U.S. Patents and Trademarks Office patent data exploration tool that disambiguates patent inventors using an entity resolution algorithm. We provide a data collection methodology and tailored performance estimators that account for sampling biases. Our approach is simple, practical and principled -- key characteristics that allow us to paint the first representative picture of PatentsView's disambiguation performance. This approach is used to inform PatentsView's users of the reliability of the data and to allow the comparison of competing disambiguation algorithms.

Since the development of writing 5000 years ago, human-generated data gets produced at an ever-increasing pace. Classical archival methods aimed at easing information retrieval. Nowadays, archiving is not enough anymore. The amount of data that gets generated daily is beyond human comprehension, and appeals for new information retrieval strategies. Instead of referencing every single data piece as in traditional archival techniques, a more relevant approach consists in understanding the overall ideas conveyed in data flows. To spot such general tendencies, a precise comprehension of the underlying data generation mechanisms is required. In the rich literature tackling this problem, the question of information interaction remains nearly unexplored. First, we investigate the frequency of such interactions. Building on recent advances made in Stochastic Block Modelling, we explore the role of interactions in several social networks. We find that interactions are rare in these datasets. Then, we wonder how interactions evolve over time. Earlier data pieces should not have an everlasting influence on ulterior data generation mechanisms. We model this using dynamic network inference advances. We conclude that interactions are brief. Finally, we design a framework that jointly models rare and brief interactions based on Dirichlet-Hawkes Processes. We argue that this new class of models fits brief and sparse interaction modelling. We conduct a large-scale application on Reddit and find that interactions play a minor role in this dataset. From a broader perspective, our work results in a collection of highly flexible models and in a rethinking of core concepts of machine learning. Consequently, we open a range of novel perspectives both in terms of real-world applications and in terms of technical contributions to machine learning.

This is the third post in a series summarising work that seeks to provide a theory of generalisation in Deep Neural Networks (DNNs). Briefly, the first post summarises evidence that DNNs trained with stochastic optimisers (like SGD) find functions with probability proportional to their volume in parameter-space, and the second post argues that these high-volume functions are'simple', thus explaining why DNNs generalise. In the following, we summarise results in [1] which explain why the'flatness of the loss landscape' has been shown to correlate with generalisation -- a well-known result (see e.g. They provide substantial empirical evidence that this correlation is actually a combination of (1) a weak correlation between the local flatness and the volume of the surrounding function, and (2) a strong correlation between volume and generalisation. This combination produces a weak correlation between'flatness' and generalisation.