We introduce the problem of distributional matrix completion: Given a sparsely observed matrix of empirical distributions, we seek to impute the true distributions associated with both observed and unobserved matrix entries. This is a generalization of traditional matrix completion where the observations per matrix entry are scalar valued. To do so, we utilize tools from optimal transport to generalize the nearest neighbors method to the distributional setting. Under a suitable latent factor model on probability distributions, we establish that our method recovers the distributions in the Wasserstein norm. We demonstrate through simulations that our method is able to (i) provide better distributional estimates for an entry compared to using observed samples for that entry alone, (ii) yield accurate estimates of distributional quantities such as standard deviation and value-at-risk, and (iii) inherently support heteroscedastic noise. We also prove novel asymptotic results for Wasserstein barycenters over one-dimensional distributions.

Consistency of case bases is a way to avoid the problem of retrieving conflicting constraining precedents for new cases to be decided. However, in legal practice the consistency requirements for case bases may not be satisfied. As pointed out in (Broughton 2019), a model of precedential constraint should take into account the hierarchical structure of the specific legal system under consideration and the temporal dimension of cases. This article continues the research initiated in (Liu et al. 2022; Di Florio et al. 2023), which established a connection between Boolean classifiers and legal case-based reasoning. On this basis, we enrich the classifier models with an organisational structure that takes into account both the hierarchy of courts and which courts issue decisions that are binding/constraining on subsequent cases. We focus on common law systems. We also introduce a temporal relation between cases. Within this enriched framework, we can formalise the notions of overruled cases and cases decided per incuriam: such cases are not to be considered binding on later cases. Finally, we show under which condition principles based on the hierarchical structure and on the temporal dimension can provide an unambiguous decision-making process for new cases in the presence of conflicting binding precedents.

This paper presents a novel nearest neighbor search algorithm achieving TPU (Google Tensor Processing Unit) peak performance, outperforming state-of-the-art GPU algorithms with similar level of recall. The design of the proposed algorithm is motivated by an accurate accelerator performance model that takes into account both the memory and instruction bottlenecks. Our algorithm comes with an analytical guarantee of recall in expectation and does not require maintaining sophisticated index data structure or tuning, making it suitable for applications with frequent updates. Our work is available in the open-source package of Jax and Tensorflow on TPU.

Nearest neighbor is a popular class of classification methods with many desirable properties. For a large data set which cannot be loaded into the memory of a single machine due to computation, communication, privacy, or ownership limitations, we consider the divide and conquer scheme: the entire data set is divided into small subsamples, on which nearest neighbor predictions are made, and then a final decision is reached by aggregating the predictions on subsamples by majority voting. We name this method the big Nearest Neighbor (bigNN) classifier, and provide its rates of convergence under minimal assumptions, in terms of both the excess risk and the classification instability, which are proven to be the same rates as the oracle nearest neighbor classifier and cannot be improved. To significantly reduce the prediction time that is required for achieving the optimal rate, we also consider the pre-training acceleration technique applied to the bigNN method, with proven convergence rate. We find that in the distributed setting, the optimal choice of the neighbor k should scale with both the total sample size and the number of partitions, and there is a theoretical upper limit for the latter.

This paper introduces an efficient and robust framework for hybrid query (HQ) processing, which combines approximate nearest neighbor search (ANNS) with attribute constraint. HQ aims to find objects that are similar to a feature vector and match some structured attributes. Existing methods handle ANNS and attribute filtering separately, leading to inefficiency and inaccuracy. Our framework, called native hybrid query (NHQ), builds a composite index based on proximity graph (PG) and applies joint pruning for HQ. We can easily adapt existing PGs to this framework for efficient HQ processing.

We consider the problem of learning the nearest neighbor graph of a dataset of n items. The metric is unknown, but we can query an oracle to obtain a noisy estimate of the distance between any pair of items. This framework applies to problem domains where one wants to learn people's preferences from responses commonly modeled as noisy distance judgments. In this paper, we propose an active algorithm to find the graph with high probability and analyze its query complexity. In contrast to existing work that forces Euclidean structure, our method is valid for general metrics, assuming only symmetry and the triangle inequality.

We introduce Exemplar VAEs, a family of generative models that bridge the gap between parametric and non-parametric, exemplar based generative models. Exemplar VAE is a variant of VAE with a non-parametric latent prior based on a Parzen window estimator. To sample from it, one first draws a random exemplar from a training set, then stochastically transforms that exemplar into a latent code and a new observation. We propose retrieval augmented training (RAT) as a way to speed up Exemplar VAE training by using approximate nearest neighbor search in the latent space to define a lower bound on log marginal likelihood. To enhance generalization, model parameters are learned using exemplar leave-one-out and subsampling.

This paper introduces SOAR: Spilling with Orthogonality-Amplified Residuals, a novel data indexing technique for approximate nearest neighbor (ANN) search. SOAR extends upon previous approaches to ANN search, such as spill trees, that utilize multiple redundant representations while partitioning the data to reduce the probability of missing a nearest neighbor during search. Rather than training and computing these redundant representations independently, however, SOAR uses an orthogonality-amplified residual loss, which optimizes each representation to compensate for cases where other representations perform poorly. This drastically improves the overall index quality, resulting in state-of-the-art ANN benchmark performance while maintaining fast indexing times and low memory consumption.

Current state-of-the-art approximate nearest neighbor search (ANNS) algorithms generate indices that must be stored in main memory for fast high-recall search. This makes them expensive and limits the size of the dataset. We present a new graph-based indexing and search system called DiskANN that can index, store, and search a billion point database on a single workstation with just 64GB RAM and an inexpensive solid-state drive (SSD). Contrary to current wisdom, we demonstrate that the SSD-based indices built by DiskANN can meet all three desiderata for large-scale ANNS: high-recall, low query latency and high density (points indexed per node). On the billion point SIFT1B bigann dataset, DiskANN serves 5000 queries a second with 3ms mean latency and 95% 1-recall@1 on a 16 core machine, where state-of-the-art billion-point ANNS algorithms with similar memory footprint like FAISS and IVFOADC G P plateau at around 50% 1-recall@1.

Nearest neighbor is a popular nonparametric method for classification and regression with many appealing properties. In the big data era, the sheer volume and spatial/temporal disparity of big data may prohibit centrally processing and storing the data. This has imposed considerable hurdle for nearest neighbor predictions since the entire training data must be memorized. One effective way to overcome this issue is the distributed learning framework. Through majority voting, the distributed nearest neighbor classifier achieves the same rate of convergence as its oracle version in terms of the regret, up to a multiplicative constant that depends solely on the data dimension.