k-Nearest Neighbors

We introduce Round-Trip KNN Clustering (RTKNNC), a graph-based method for finding cluster structure at several neighbourhood scales without requiring the number of clusters in advance. Unlike approaches that first make a $k$-nearest-neighbour (KNN) graph undirected, RTKNNC keeps both directions of the neighbour relation: which points a given point selects and which points select it. Incoming selections are treated as weighted votes that help decide which local connections remain visible during a recursive forward-and-reverse traversal. Repeating the procedure for increasing $K$ reveals how groups persist or merge as the neighbourhood scale grows; for the reference inverse-square model before structural refinement, clusters can merge but do not split. Because graph connectivity can occasionally join distinct groups through a sparse bridge or a small region of overlap, we add an optional label-free refinement. It first tests whether an already formed component is better described by two or three Gaussian subpopulations, and accepts a subdivision only when the proposed groups are large enough and consistent with the visible KNN graph. Across eight synthetic datasets and $K=2,\ldots,16$, independent C and Python implementations produced identical partitions in all 120 reference runs. Refinement increased adjusted Rand index from $0.7817$ to $0.9627$ on a variable-density benchmark and from $0.8083$ to $0.9853$ on a sparse-bridge benchmark. Comparisons with seven external clustering methods show competitive performance while preserving a label-free cluster-construction process.

In this paper, a methodology to design fair graph convolutional neural networks (GCNs) is developed and tested over several application data sets. The graphs that are used as inputs to the network are constructed by a k-nearest neighbor-based preprocessing procedure, while fairness issues are considered in terms of the equalized odds criterion. To effectively incorporate the above heterogenous information, the equalized odds criterion is directly embedded into the model's optimization objective through an additional fairness-driven loss functional term. The proposed methodology investigates how varying the neighborhood size in the k-NN algorithm during graph construction influences both the classification performance and the fairness of the resulting models. Extensive experimentation is conducted on three real-world tabular datasets with known biases, evaluating the interplay between graph structure and fairness enforcement. The results demonstrate that the choice of the value of the parameter k critically impacts the performance trends, either steadily improving or peaking at intermediate values depending on dataset characteristics, while the application of fairness constraints significantly mitigates disparities in false positive and false negative rates across groups defined by the protected variable at hand, without incurring major sacrifices in overall accuracy. This study highlights the importance of jointly optimizing the graph construction process and fairness objectives in GCN-based learning, providing a systematic approach toward building more equitable and effective graph-based models.

We extend recent work establishing an equivalence between one-layer transformers and nearest-neighbor classifiers in the binary setting to the multiclass case. By leveraging the simplex encoding, we show that one-layer transformers with an argmax classification head behave identically to a one-nearest-neighbor classifier in the multiclass setting. This closes a gap left by prior work, whose multiclass result relied on a non-standard rounding-based approach rather than the typical argmax head used in practice.

The $k$-Nearest Neighbor~(KNN) algorithm is widely used across various tasks. The selection of the $k$ value is a key issue because it significantly impacts performance. In this paper, an adaptive and efficient KNN approach via granular-ball computing is proposed. The method consists of two stages. \textcolor{black}{In the training stage, the dataset is first coarsely partitioned to reduce the complexity of data distributions within a granular ball, and then the Fisher criterion is introduced to control ball splitting and stopping, yielding a multi-granularity granular ball representation. In the prediction stage, the nearest granular ball is first located through a weighted distance mechanism, and an adaptive neighborhood is then constructed around the test sample. The effective $k$ value is dynamically determined by the actual number of samples contained in this neighborhood. The neighborhood induced by the nearest granular ball provides more stable local group information, thereby improving robustness against noise and local perturbations.} Experimental results demonstrate that the proposed method outperforms existing KNN variants across multiple datasets in terms of both accuracy and efficiency. The code has been open-sourced for reproducibility: https://github.com/lianxiaoyu724/Adaptive-GBKNN.

A trained molecular property model can be refined at test time by correcting each prediction with the measured labels of the most similar training molecules, a retraining-free procedure we call neighbor fusion; evidential neural networks make it principled by using their aleatoric and epistemic uncertainty to parameterize a Bayesian update. Our main contribution, PG-EVIKAL, learns a property-distance metric to re-rank structurally similar neighbors by their property relevance before fusion, building on EVIKAL (scalar Kalman filter) and GP-EVIKAL (Gaussian process variant handling correlated neighbors). Evaluated on 16 molecular datasets, PG-EVIKAL reduces RMSE relative to the evidential model baseline on 14 of them, with a median reduction of 19.4%, and improves calibration; in sequential-assay scenarios it further incorporates newly measured molecules, refining predictions as they arrive without retraining. This work demonstrates that evidential uncertainty decomposition is not merely a calibration objective but an actionable inference resource that enables test-time refinement of molecular property predictions.

Data Shapley is the standard principled answer to which training points are worth what, and its k-nearest-neighbor (KNN) specialization is the version deployed in practice: the exact estimator shipped by toolkits such as pyDVL and OpenDataVal. Exact algorithms are known for unweighted KNN and for weighted KNN classification, but weighted KNN regression and soft-label prediction have resisted: the only exact method is an O(N^K) brute force, exponential in neighborhood size K. The obstruction: the weighted regression prediction is a ratio of two coalition-dependent sums, whose normalization denominator breaks the additive, threshold, and duplication structures the prior polynomial algorithms rely on. We close this gap. We give (i) the first pseudo-polynomial-time exact algorithm (polynomial in N and K at fixed lattice precision) for weighted KNN-regression Data Shapley, a counting dynamic program over the joint integer state (sum of w, sum of w*y), verified against exhaustive enumeration with zero mismatch on 12,716 adversarial instances; (ii) a certified FPTAS for continuous weights and targets, with a machine-checkable per-value error certificate never violated across 86,400 checks; (iii) a complexity landscape, including an unconditional Omega(D_w) output-size lower bound and access-model hardness results; and (iv) a weighted soft-label multi-class extension. We release an open-source, CPU-only library and the first exact weighted-regression Data Shapley ground truth. On downstream mislabel detection our exact values are statistically equivalent to Monte-Carlo Data Shapley (dataset-level TOST, n=8, p<10^-4), the pre-registered outcome; the value of exactness is instead determinism, a certified error bound, and an exact reference for auditing estimators: Monte-Carlo did not reproduce the exact top-10% ranking at any budget tested, up to 3,000 permutations (~1.28e6 utility evaluations).

Convergence-rate analysis for classifiers is often conducted under either Tsybakov margin or Massart margin. The former is a relatively weak condition that typically yields polynomial rates, while the latter is substantially stronger but can guarantee exponential rates. In this paper, we introduce a new condition, called Boltzmann margin, that bridges the gap between these two regimes. It is weaker than Massart margin, generally stronger than Tsybakov margin, and can imply many of their properties under suitable conditions. We apply Boltzmann margin to the analysis of kNN classifiers and establish the first near-exponential convergence rates for kNN classification. We also present extensions of the main results and provide numerical evidence supporting the main theoretical implications.

The Gromov--Wasserstein (GW) distance provides a framework for comparing metric measure spaces, regardless of their underlying structure or geometry. For network-based data, it enables direct comparisons of graphs with different numbers of nodes, without requiring an embedding or other abstraction. Furthermore, through a variant of GW known as fused Gromov--Wasserstein (fGW), it is also possible to incorporate node features in addition to graph structure. In this work, we implement $k$-nearest neighbors ($k$-NN) classification using the GW and fGW distances. We prove the universal consistency of the GW-$k$-NN classifier on the space of equivalence classes of metric measure spaces with finite support and uniform probability measure. By viewing graphs as finitely supported metric measure spaces equipped with the pairwise distance metric and a uniform probability measure on the nodes, we obtain universal consistency of GW-$k$-NN for the space of graphs. Likewise for fGW-$k$-NN, we prove universal consistency on the space of weak isomorphism classes of structured objects consisting of metric measure spaces with finite support and uniform probability measure and feature maps into Euclidean space, thus establishing universal consistency on the space of node-attributed graphs. Our numerical experiments show that GW-$k$-NN and fGW-$k$-NN consistently perform well across multiple graph datasets, suggesting that metric classifiers such as $k$-NN work well in the GW framework.

AI efficiency at scale is becoming critical in finance as market data volumes surge across equities, ETFs, FX, options, and high-frequency trading streams. This growth creates a core challenge for mature financial AI systems: models must learn from larger historical corpora while still meeting real-time latency constraints in trading, risk management, and derivative pricing. We use exact nearest-neighbor learning for high-frequency financial time series as a concrete case study to show that Mojo-based financial AI can address this challenge. We introduce a Mojo SIMD k-d tree with variance-based splitting, contiguous flat-buffer storage, and compile-time vectorized distance computation. We also provide a runtime result showing that, under standard pruning and implementation-cost assumptions, the Mojo SIMD k-d tree asymptotically dominates Mojo SIMD brute force and scikit-learn's k-d tree in the fixed-stock, large-$n$, moderate-dimensional regime. Empirically, across eight financial datasets on x86 and ARM64 with up to 277K training samples, the method achieves 17.5--21.6$\times$ speedup over scikit-learn's k-d tree on x86 and 28.1--43.5$\times$ over scikit-learn brute force on ARM64 equity/ETF datasets, while preserving exact outputs. Beyond nearest-neighbor inference, Mojo's compiled execution enables an Extra Trees-based implied-volatility pricing model to train on $10\times$ more options data, reducing put-IV RMSE by 8.0%. These results position Mojo as a scalable, production-ready stack for financial AI and a promising foundation for efficient AI in other data-intensive fields. \keywords{Financial AI \and AI Efficiency \and Mojo \and SIMD \and K-D Trees \and KNN \and High-Frequency Trading \and Financial Time Series \and Scaling}

The shift from Convolutional Neural Networks to Transformers has reshaped computer vision, yet these two architectural families are typically viewed as fundamentally distinct. Convolutional Neural Networks are defined by spatially local convolution operations, while Transformers rely on global self-attention. We argue that convolution and self-attention, despite their apparent differences, can be unified within a single k-nearest neighbor aggregation framework. The critical insight is that both operations are special cases of neighbor selection and weighted aggregation. Convolution selects neighbors by spatial proximity while self-attention selects by feature similarity, revealing that they lie on a continuous spectrum rather than representing categorically different computations. We introduce Convolutional Nearest Neighbors (ConvNN), a unified framework that formalizes this connection. ConvNN exactly recovers standard and depthwise convolution by restricting neighbor selection to normalized spatial coordinates, and exactly recovers self-attention and its sparse variants, including KVT-attention, by replacing spatial proximity with scaled dot-product similarity. Beyond these special cases, ConvNN serves as a drop-in replacement for both convolution and attention layers, enabling systematic exploration of the intermediate spectrum between local and global aggregation through configurable similarity functions, neighbor selection strategies, positional encodings, and aggregation kernels.

Nearest-neighbor methods are fundamental to classical and modern machine learning, yet their geometric properties are typically analyzed under independent sampling. In this paper, we study the nearest-neighbor radii under dependent sampling. We consider strong mixing dependent observations and ask whether dependence changes the scale of nearest-neighbor neighborhoods. We establish distribution-free almost sure convergence under polynomial mixing and sharp non-asymptotic moment bounds under geometric mixing. The moment bounds depend on the local intrinsic dimension rather than the ambient dimension, making the results applicable to high-dimensional data concentrated near lower-dimensional manifolds. Synthetic experiments and real-world time-series benchmarks support the theory, showing that nearest-neighbor geometry remains informative under dependence sampling.

The performance of machine learning models often relies on large labeled datasets; however, data collected from diverse sources can contain label noise. Recent work has shown that, in noisy settings, there may exist a subset of the training data on which models can achieve performance comparable to training on a noise-free dataset. A widely used method for identifying such subsets is cutstats, which employs k-nearest neighbors (k-NN) to detect low-noise samples. However, its performance on high-dimensional data remains largely unexplored. In this work, we formally establish that the performance of a classifier trained on a subset of a noisy dataset selected via cutstats is influenced by the accuracy of k-NN. We further demonstrate that, in noisy environments, exploiting data invariance and knowledge of underlying symmetries can significantly enhance the performance of k-NN, bringing it closer to the Bayes optimal classifier even in high-dimensional regimes. Finally, we show that for real-world scenarios, where information about the underlying invariance is only partially known, learnt invariant representations can still facilitate the identification of near-optimal subsets.

Semi-supervised classification, where unlabeled data are massive but labeled data are limited, often arises in machine learning applications. We address this challenge under high-dimensional data by leveraging the manifold and cluster assumptions. Based on the Fermat distance, a density-sensitive metric that naturally encodes the cluster assumption, we propose the weighted $k$-nearest neighbors (NN) classifier and multidimensional scaling (MDS)-induced classifiers. The use of MDS with a large target dimension allows the effective application of linear classifiers to complex manifold data. Theoretically, we derive a sharp lower bound for the expected excess risk within clusters and prove that the weighted $k$-NN classifier utilizing the true Fermat distance is minimax optimal. Furthermore, we explicitly quantify the utility of unlabeled data by showing that the error arising from estimating the Fermat distance decays exponentially with the pooled sample size. Such a rate is much faster than the related rates in the literature. Extensive experiments on synthetic and real datasets demonstrate competitive or superior performance of our approaches compared to state-of-the-art graph-based semi-supervised classifiers.

The development of 3D scanning technology has enabled the acquisition of massive point cloud models with diverse structures and large scales, thereby presenting significant challenges in point cloud processing. Fast neighboring points search is one of the most common problems, which is frequently used in model reconstruction, classification, retrieval and feature visualization. Hash function is well known for its high-speed and accurate performance in searching high-dimensional data, which is also the core of the proposed 2L-LSH. Specifically, the 2L-LSH algorithm adopts a two-step hash function strategy, in which the popular step divides the bounding box of the point cloud model and the second step constructs a generalized table-based data structure. The proposed 2L-LSH offers a highly efficient and accurate solution for fast neighboring points search in large-scale 3D point cloud models, making it a promising technique for various applications in the field. The proposed algorithm is compared with the well-known methods including Kd-tree and Octree; the obtained results demonstrated that the proposed method outperforms Kd-tree and Octree in terms of speed, i.e. the time consumption of kNN search can be 51.111% and 94.159% lower than Kd-tree and Octree, respectively. And the RN search time can be 54.519% and 41.840% lower than Kd-tree and Octree, respectively.

Fixed Radius Near Neighbor (FRNN) search is an alternative to the widely used k Nearest Neighbors (kNN) search. Unlike kNN, FRNN determines a label or an estimate for a test sample based on all training samples within a predefined distance. While this approach is beneficial in certain scenarios, assuming a fixed maximum distance for all training samples can decrease the accuracy of the FRNN. Therefore, in this paper we propose the Adaptive Radius Near Neighbor (ARNN) and the Weighted ARNN (WARNN), which employ adaptive distances and in latter case weights. All three methods are compared to kNN and twelve of its variants for a regression problem, namely WiFi fingerprinting indoor positioning, using 22 different datasets to provide a comprehensive analysis. While the performances of the tested FRNN and ARNN versions were amongst the worse, three of the four best methods in the test were WARNN versions, indicating that using weights together with adaptive distances achieves performance comparable or even better than kNN variants.

Convolutional Neural Networks and Vision Transformers are the two dominant architectural families in computer vision, defined by spatially local convolution and global self-attention respectively. Despite their apparent differences, we show that both operations are special cases of a single $k$-nearest neighbor aggregation framework: convolution selects neighbors by spatial proximity while attention selects by feature similarity, placing them at two ends of a shared operational spectrum. We introduce Convolutional Nearest Neighbors (ConvNN), a unified framework that exactly recovers standard and depthwise convolution, self-attention, and sparse attention variants including KVT-attention as special cases, and exposes the design space of neighbor-selection strategies between them through configurable similarity functions, positional encodings, and aggregation kernels. We validate ConvNN on ImageNet-1K classification across two complementary architectures: a hybrid branching layer in ResNet-50 that combines local and global feature learning, improving top-1 accuracy by 3.0% over the ResNet-50 baseline, and ConvNN-attention in ViT-Base that achieves 81.64% top-1 accuracy, surpassing standard multi-head self-attention by 0.7%. Together, these results demonstrate that ConvNN provides a principled foundation for designing operations that bridge convolutional and attention-based computation.

This paper introduces a framework for uncertainty quantification in regression models defined on metric spaces. Using a proposed notion of homoscedasticity, we define a conformal prediction algorithm that provides finite-sample marginal coverage guarantees and fast convergence rates to the oracle prediction region. For heteroscedastic settings, we introduce a kNN procedure that yields locally adaptive prediction radii in general metric spaces. Although this procedure does not provide the same finite-sample guarantees as the conformal algorithm, it is designed to improve local coverage calibration without imposing smoothing assumptions. Both procedures are compatible with a broad range of regression algorithms and scale to large datasets, allowing practitioners to use their preferred models and incorporate domain-specific knowledge. Building on the heteroscedastic $k$NN approach, we also develop a flexible sequential extension for metric-space-valued time series based on nearest-neighbor expert aggregation. We establish the consistency of the proposed estimators under minimal conditions. Finally, we illustrate the practical utility of our framework in personalized medicine applications involving random objects such as probability distributions and graph Laplacians.