Sample Size

Sample size determination for machine learning (ML) prediction models is challenging because conventional power analysis typically requires the predictor-outcome relationship and effect structure to be specified a priori. Nonlinear ML models learn complex prediction surfaces that do not admit straightforward analytical power calculations. We propose a framework that approximates nonlinear ML models with localized linear representations and estimates sample size requirements by evaluating statistical power across these local regions.

The small-sample-size (SSS) problem remains a fundamental challenge in machine learning when labeled data are scarce due to cost, accessibility, or ethical constraints. While numerous approaches have been proposed, existing methods often struggle to maintain stable and discriminative representations under high-dimensional and limited-data conditions. Kernelized Linear Principal Component Discriminant Analysis (KLPCDA), a recently proposed modular framework, integrates variance preservation, inter-class separability, and intra-class compactness within a unified kernel space. Although its formulation has shown promising initial results, a systematic understanding of how its components interact across diverse SSS scenarios remains lacking. In this paper, we present a systematic cross-domain study of KLPCDA to characterize the interaction mechanisms among its core objectives. We analyze the behavior of its seven variants across multiple real-world SSS tasks, including hyperspectral image classification, mechanical fault diagnosis, medical diagnosis, and face recognition. Through extensive experiments and ablation studies, we investigate how different objective combinations influence performance under varying conditions such as noise, class imbalance, and high dimensionality. Our analysis reveals consistent patterns in the interaction of the three core objectives variance, between-class, and within-class terms, providing a unified and interpretable understanding of their roles in stabilizing representations and enhancing discrimination in SSS settings. Based on these findings, we further derive practical guidelines for selecting appropriate KLPCDA variants under different data characteristics. Experimental results demonstrate that KLPCDA achieves strong and robust performance across domains, while maintaining low computational complexity suitable for resource-constrained environments.

Recent advancements in transformer length generalization theory enable us to reliably predict when a transformer can learn to solve a task. In particular, the C-RASP hypothesis (a formalized version of the so-called RASP-l conjecture) posits that transformers length-generalize on a task if and only if a solution is expressible in the C-RASP language. While this hypothesis has strong empirical validation, theoretical problems arise from the fact that no computable length generalization bounds exist for C-RASP, alongside the discovery of seemingly contradictory experiments. To address these problems, we refine the C-RASP hypothesis utilizing the recently-proposed fragments C-RASP+ and C-RASP1. These fragments have computable length generalization bounds, though in the worst case requiring an extremely large (double exponential) sample size. It is an open question whether these sample size bounds are tight. In this paper, we resolve this open question by providing an exponentially tighter bound. In doing so, we show a polynomial length generalization bound for transformers if we adopt compressed strings, via a novel connection to power words. As an application, we show how this yields a fine-grained analysis of the C-RASP conjecture that resolves contradicting experimental evidence against it.

Low Power Wide Area Networks like LoRa are increasingly deployed for smart city applications, requiring accurate path loss prediction for effective network planning. Traditional (empirical) propagation models often exhibit limited accuracy in these scenarios. We investigate machine learning models for LoRa path loss prediction, systematically analyzing how prediction accuracy scales with training set size using real-world measurements from an urban deployment. Our approach employs a Random Forest with LiDAR-derived terrain features and k-Nearest Neighbors with coordinate data, comparing their performance against established empirical models and specialized LPWAN models. Under random pooled splits, both ML models consistently outperform the considered baseline models across the evaluated training-set sizes. At maximum training size, they achieve RMSE values below 6.5 dB compared to 9.7 dB for the best baseline, indicating accurate within-deployment interpolation. A leave-one-gateway-out check qualifies this result: RF shows placement-dependent transfer to held-out gateways, with moderate degradation for several gateways but larger errors for others, whereas coordinate-only k-NN degrades substantially when the gateway location is unseen

Randomized experiments are often run in one population to guide decisions in another. Allocating by experimental proportions wastes budget on groups that rarely appear in deployment, whereas allocating by deployment proportions under-samples groups that are hard to measure precisely. We propose \textbf{TWNA} (Target-Weighted Neyman Allocation), a two-stage stratified design that uses pilot estimates of group--arm outcome variances to allocate final-stage sample sizes and treatment probabilities for target-weighted group average treatment effect (GATE) precision. The oracle rule has a closed form and balances deployment importance with statistical difficulty; the plug-in rule recovers it as pilot variance estimates stabilize. We also extend TWNA to handle uncertainty about deployment composition, remaining robust whether the target mix is roughly known or entirely unknown. Finally, we distinguish this weight robustness from a pilot-robust variant for skewed, rare-event, or contaminated outcomes. Simulations and real-covariate benchmarks show the largest gains when groups are both deployment-important and difficult to measure.

Per-subgroup fairness audits of medical image classifiers face a sample-size problem: minority subgroups in held-out test sets have so few samples that the resulting confidence intervals on per-subgroup performance are wider than the bias the audit is meant to detect. We argue that a demographically-conditioned synthetic generator can do both: mitigate bias on the training side and detect bias on the evaluation side. Working on COVID-19 chest CT classification with an end-to-end fine-tuned Stable Diffusion 2.1 generator, we make two findings. For bias mitigation (training), a demographically-balanced synthetic cohort is most useful as a pretraining prior, not as joint augmentation: with the same fixed data, sequential pretraining followed by fine-tuning substantially outperforms joint augmentation, and the resulting classifier surpasses the full-real baseline at \sim$$100\times real-data efficiency. For bias detection (evaluation), across five synthetic minority cohorts and five classifier seeds, the synthetic estimator reproduces the subgroup ranking of a well-powered real oracle (Spearman $ρ= 1.00$ on MCC and Recall) and gives the more reliable per-cell estimate where the small real test set runs out of samples. The synthetic cohort is therefore most useful in exactly the cells that fairness audits care about, as both a fix for and a measure of subgroup bias.

Weighted empirical measures on compact manifolds arise in importance sampling, particle approximations, posterior summaries, quadrature, and representation learning. Standard weight-only summaries, such as ordinary effective sample size, ignore the geometry of the support. We introduce heat-kernel entropy profiles, a multiscale summary that diffuses weighted atoms by intrinsic heat flow and tracks nonuniformity across scales. For order-two Rényi entropy, the profile is computable from pairwise heat-kernel overlaps and yields a geometric effective sample size that discounts nearby or duplicate particles while matching ordinary effective sample size for well-separated particles. We prove monotonicity, small- and large-scale asymptotics, deterministic-weight consistency, and a bounded-ratio self-normalized importance-sampling extension for compact manifolds without boundary. On spheres, the unlogged profile decomposes into spherical-harmonic energies that recover mean-direction, von Mises-Fisher-type, and Bingham-type summaries. Sphere-based experiments show that the profile reveals antipodal, girdle, multimodal, and duplicate-particle structure missed by weight-only and first-moment spherical summaries.

Deep learning models for surface electromyography (sEMG) can benefit substantially from subject-specific (re-)calibration, since no sufficiently large and diverse datasets are available to train fully generic decoders. However, for user acceptance, the number of repetitions that can realistically be collected during calibration is severely limited, which increases the risk of overfitting and, in extreme cases, can even degrade performance compared to the uncalibrated model. Classical overfitting indicators such as validation performance and regularization with early stopping are difficult to apply in this low-sample regime, as they require additional held-out data that is rarely available in practical calibration scenarios. In this work, we investigate a recently proposed class of memorization indicators based solely on the activation statistics of rectified linear units (ReLU) in deep neural networks, which can be computed directly from training data without any extra validation set. We conduct a transferlearning experiment on a benchmark sEMG dataset, where a convolutional neural network is first pre-trained on multiple subjects and subsequently fine-tuned on individual users using only a small number of repetitions. During calibration, we monitor both decoding performance and the activation behaviour of the last hidden layer. Our results provide first evidence that decreases in test accuracy during fine-tuning are ac companied by characteristic changes in activation rates, indicating that activation-based memorization indicators are a promising tool for early spotting of unsuccessful learning in low-sample sEMG calibration settings.

Researchers increasingly use text classification--supervised models or large language models--to measure constructs from natural language, providing metrics such as recall and precision as evidence of their validity. Yet, though these metrics are point estimates subject to sampling variation, measures of uncertainty are inconsistently reported alongside them. Further, when they are reported, they are often estimated with methods that are not appropriate when relevant labelled datasets are small or performance is high. To increase and improve confidence interval reporting in the field, this paper evaluates confidence interval methods for performance metrics under conditions typical of social science text classification: small to moderate sample sizes, infrequent constructs, and texts nested within individuals. Across simulations, default methods such as the Wald interval and the basic percentile bootstrap are the least accurate, with coverage sometimes far below the nominal 95% level. Accuracy is improved with the use of Agresti-Coull, Wilson, Clopper-Pearson, and a novel pseudo-count regularized bootstrap (which is particularly relevant to the calculation of F1). When texts are nested within individuals, we demonstrate that adjustment for both effective N and the appropriate degrees of freedom is necessary for producing accurate analytic intervals. Among bootstrap intervals, the hierarchical bootstrap is more accurate than the cluster bootstrap when individuals produce a moderate number of texts but overly conservative when individuals produce only a few. By providing guidance to the field on appropriate interval estimation, we aim to improve the transparency of machine learning applications, and to encourage greater attention to the validation sample size at the design stage.

We show every multi-group learner in the transductive setting may incur a multiplicative penalty in its error rate on some group relative to the error rate achievable in the single-group setting, and the penalty can increasing linearly with the number of groups, up to roughly the square-root of the sample size. This stands in stark contrast to optimal multi-group learners in an analogous (group-realizable) statistical setting, where the penalty is always at most logarithmic in the sample size and independent of the number of groups.

Active learning is now standard practice in labeling ecological data, enabling ecologists to quickly process large volumes of field data to understand and monitor natural environments. Current practices evaluate active learning inductively, estimating predictive performance on a held-out test set. We argue that this evaluation is misaligned with most ecological tasks, where the goal is to transductively label an entire pool of data as efficiently as possible. We demonstrate that ignoring the human-in-the-loop underestimates the importance of continuing to label, particularly for classes in the long tail which may be of disproportionate ecological importance (rare species, uncommon behaviors, etc.). Our analysis shows that, for this long tail, the transductive objective shifts importance from prediction to discovery: the true challenge becomes finding "needles in the haystack," examples of rare classes that are embedded within dense regions of abundant classes in the latent geometry, which we quantify with a novel metric of sampling difficulty. Finally, to translate these insights to practical ecological workflows, we propose a conservative hybrid stopping criterion inspired by ecological rarefaction curves, and show that combining predictive performance with discovery criteria reduces premature stopping on long-tailed pools, improving rare-class recovery when discovery, not classification, is the limiting factor.

African languages have very little labelled data, and it is unclear if augmenting the quantity of annotation data reliably enhances downstream performance. The study is a systematic sample-size scaling study of natural language inference (NLI) on 16 African languages based on the AfriXNLI benchmark. Under controlled conditions, two multilingual transformer models with roughly 0.6B parameters XLM-R Large fine-tuned on XNLI and AfroXLM-R Large are tested on sample sizes of between 50 and 500 labeled examples and average their results across random subsampling runs. As opposed to the usual belief of monotonic increase with increased data, we find a strongly language sensitive and often non-monotonic scaling behavior. Some languages show early saturation or decrease in performance with sample size as well as high variance in low resource regimes. These results indicate that the volume of data is not enough to guarantee stable profits to African NLI, creating the necessity of language sensitive datasets creation and stronger multi-lingual modelling strategies.

Pre-training data mixtures are commonly tuned by running small-scale experiments and extrapolating to the target training budget. When high-quality data is scarce and must be repeated, this extrapolation frequently fails, but the source of the failure has not been isolated. We show that a primary culprit is a repetition mismatch: because high-quality datasets are small, their repetition rate changes as the training budget grows, shifting the optimal mixture in ways that small-scale proxy experiments do not anticipate. A subsampling procedure that matches the target repetition rate controls for this effect. In a two-source setting combining limited high-quality data with web crawl, a single repetition-controlled experiment using only 1/16 of the target tokens recovers a mixture within 0.10 of the optimum on Wiki-Text for a 1.17B parameter model, compared to an error of 0.85 without repetition control. Achieving comparable accuracy without repetition control requires multiple training horizons, consuming 19%, 44%, and 94% of the target token budget when using the results from two, three, and four horizons respectively. With three data sources, the larger mixture space requires more than a single experiment to constrain, but the approach remains effective: at the 757M scale, just two repetition-controlled horizons recover the optimal mixture, outperforming baselines that instead require the full two-source experiments to construct. Our results reveal that repetition dynamics, not scale alone, shape whether small-scale mixture experiments generalize. More broadly, they suggest that data repetition deserves treatment as a first-class variable in mixture optimization, rather than an inconvenient side effect of limited data.

Differential privacy changes the effective sample size governing CVaR learning. For tail mass $τ$, the privacy-relevant sample size is not $n$, but $nτ$; equivalently, the effective private tail sample size is $εnτ$. Private CVaR excess risk decomposes into ordinary tail-risk statistical error and a privacy price. This decomposition is complete for scalar estimation and finite classes: scalar estimation has rate $Θ(B \min\{1,(nτ)^{-1/2}+(εnτ)^{-1}\})$, and finite classes of size $M$ have rate $Θ(B \min\{1,\sqrt{\log(2M)/(nτ)}+\log(2M)/(εnτ)\})$. These complete rates hold under pure DP, and their lower bounds extend to approximate DP in the stated small-$δ$ regimes. For convex Lipschitz learning, modular upper and lower reductions show that the CVaR-specific privacy term necessarily scales as $1/(εnτ)$, with dimension dependence inherited from private stochastic convex optimization. Together, these results identify ordinary private learning on $Θ(nτ)$ informative tail records as the canonical hard subproblem inside private CVaR learning.

Training a language model on data scattered across bandwidth-limited nodes that cannot be centralized is a setting that arises in clinical networks, enterprise knowledge bases, and scientific consortia. We study the regime in which data must remain distributed across nodes, and ask what statistical guarantees are in principle achievable under explicit bandwidth budgets; we aim to characterize what is provably possible, not to demonstrate a deployment-ready system. Existing theory treats either training-time consistency or inference-time calibration in isolation, and no prior work makes bandwidth a first-class statistical parameter. We analyze two protocols, Federated Probe-Logit Distillation (FPLD) for training and Federated Conformal RAG (FC-RAG) for inference, as the analytical vehicles for our results. Our first main result is an explicit high-probability KL-consistency rate for FPLD with simultaneous dependence on node count $K$, per-node sample size $n$, quantization budget $B$, probe-set size $m$, and vocabulary size $V$; bandwidth enters only through an exponentially vanishing quantization term. Our second main result is a distribution-free marginal-coverage bound for FC-RAG, whose novel retrieval-bandwidth slack $Δ_{\mathrm{RAG}} = f_{\max}\sqrt{K^{-2}\sum_i v(B_i)}$ makes per-node retrieval bandwidth a first-class statistical parameter, with arithmetic aggregation across $K$ nodes shrinking the slack as $K^{-1/2}$ in the per-node-uniform regime. A Pinsker-type corollary composes the two bounds into an end-to-end coverage guarantee. Synthetic experiments verify the predicted scaling along the bounds' parameters; small-scale experiments on a GPT-2 testbed illustrate that the qualitative bandwidth-accuracy tradeoff survives on a real language model. A deployment-scale empirical evaluation is out of scope.

If a person can solve a task, can measuring their brain make it easier to train a model to solve that task too? Recent NeuroAI work suggests that supplementing task training with neural recordings can modestly improve model performance and robustness. However, it is unclear when there should be a benefit from using neural data and how much benefit to expect. We formulate this question mathematically, and begin to address it theoretically using a simple, analytically tractable linear gaussian model of task targets and neural recordings. For a multimodal estimator trained on both brain data and task labels, we derive scaling laws for how performance scales with the numbers of brain and task samples. From these laws we derive relative value and exchange rates between brain samples and task samples, quantifying how much extra task samples neural data is worth as a function of task-brain alignment, neural and task noise, latent dimension, and brain data sample size. We also analyze test distribution shift, to identify conditions where brain-regularized learning can produce substantial robustness gains through learned invariances. Finally, under a fixed collection budget, we characterize the regimes in which brain data is worth collecting. Our results provide a foundation for understanding how valuable brain data could be for improving machine learning.

A Mamba state-space model trained only for next-step prediction appears to recover Granger-causal structure through a simple readout $S = |W_{out} W_{in}|$, with early experiments suggesting the phenomenon generalized across architectures and benefited from interventional data at $p < 10^{-5}$. We package the protocol used to test that claim -- standardized synthetic generators (VAR/Lorenz/CauseMe-style), three intervention semantics ($do(X=c)$, soft-noise, random-forcing), edge-provenance cards on three real datasets, and size-matched control arms -- as a reusable falsification benchmark, and walk the claim through it in five stages. The method-level claim does not survive: (i) a plain linear bottleneck does as well or better; (ii) tuned Lasso beats the bottleneck on synthetic CauseMe-style benchmarks, and on Lorenz-96 (the only real benchmark with unambiguous ground truth) classical PCMCI and Granger lead a tight cluster in which the bottleneck trails; (iii) the headline intervention advantage is roughly 60% a sample-size confound, and the residual disappears under standard $do(X=c)$ interventions, surviving only under a non-standard random-forcing scheme; (iv) even that residual reproduces, with a larger effect, in classical bivariate Granger -- the effect is method-agnostic. What survives is a narrow characterization result; the benchmark is the lasting artifact, and each stage above is one of its control arms.

Machine Learning (ML) models trained on complex health surveys such as the National Health and Nutrition Examination Survey (NHANES) often ignore primary sampling units, stratification variables, and sampling weights. This practice violates the independence assumptions of standard evaluation methods. As a result, estimates become biased, uncertainty is underestimated, and fairness assessments fail to reflect population-level disparities. We propose Survey-aware Machine Learning (SaML), a nine-step guideline that incorporates survey design metadata across the ML lifecycle. Through a scoping review of 16 methodological papers, we summarize existing work on weighted model training, design-based cross-validation, and survey-adjusted performance evaluation. We also identify gaps in hyperparameter tuning and deployment. We provide task-specific guidance that clarifies which steps are required for different analytical objectives. SaML provides a checklist for valid population inference from survey data.

Effective sample size is a standard summary of Markov chain Monte Carlo output, but it is usually attached to scalar or Euclidean summaries chosen by the analyst. For manifold-valued samples this choice is not canonical: coordinate-wise effective sample sizes can change under rotations, chart changes, or alternative embeddings of the same underlying path. We propose an intrinsic effective sample size based on kernel discrepancy. The proposed quantity is the number of independent draws that would yield the same expected squared kernel discrepancy between the empirical distribution and the target distribution. This gives an exact finite-sample risk interpretation, an asymptotic integrated-autocorrelation representation, and a coordinate-free diagnostic whenever the kernel respects the geometry of the state space. We establish invariance under transported kernels, operator and principal-direction interpretations, and consistency of a lag-window estimator under boundedness and absolute-regularity conditions. We also discuss valid kernel constructions on manifolds, emphasizing that geodesic Gaussian kernels are not generally positive definite on curved spaces. Sphere experiments illustrate rotation invariance and calibration of the proposed diagnostic against empirical distributional error.

Large Language Models (LLMs) have rapidly improved in performance across code-related tasks, making their integration into Register Transfer Level (RTL) development increasingly attractive. Mimicking the behavior of inline code assistants, many benchmarks evaluate LLMs' capabilities in code completion, either assessing the generation of entire hardware modules or the completion of a single line within a module. However both of these approaches lack the ability to control the granularity of the code-completion sample size and the syntactic range of completions. To overcome these limitations, we present a framework for language-agnostic rule completion (RuC), a grammar-driven, rule-selectable benchmark generator that automatically produces RTL code-completion tasks from a set of input hardware description sources. RuC uses the target Hardware Description Language (HDL) grammar to mask syntactically defined code regions and prompts a model to regenerate them using the surrounding unmasked code as context, enabling a controlled and scalable evaluation of the domain-specific model's code-understanding capabilities, ranging from assignments to the reconstruction of entire logic blocks. We use RuC to generate two SystemVerilog rule-completion benchmarks from the Tiny Tapeout shuttle TT07 and the CVE2 RISC-V core to demonstrate RuC's applicability to a broad range of designs, and conduct a comparative study of the code completion capabilities of modern open-source LLMs across diverse settings. Results indicate that completion performance strongly depends on the model type, the grammatical structure of the masked region, and the prompting strategy. Specifically, the highest scores are obtained with Fill-in-the-Middle (FIM) prompting. These findings highlight the value of grammar-driven, arbitrarily granular benchmarks for meaningful evaluation of LLM capabilities in RTL development workflows.

Standard evaluations of Bayesian deep learning methods assume that metric estimates are reliable, but we show this assumption fails under data scarcity. Method rankings are not only unreliable at small $n$, but also dataset-dependent in ways that point estimates cannot reveal: the same method comparison yields $P(\mathrm{MCD} \prec \mathrm{Ensemble}) = 1.000$ at $n = 50$ on one dataset and remains below $0.95$ even at $n = 500$ on another. Across the datasets we consider, no universal sample size threshold exists, which is precisely why dataset-specific posterior inference is necessary. To address this, we use a Bayesian hierarchical model with method-specific variances to treat evaluation metrics as random variables across data realizations, and we use a predictive Minimum Detectable Difference curve to assess whether an observed gap would be detectable at a given training size. Across six Bayesian deep learning methods and five regression datasets, our results show that uncertainty-aware evaluation is necessary in low-data settings, because current evidence for method superiority and predictive detectability at the same training size can diverge substantially. Our framework provides practitioners with principled tools to determine whether their evaluation data is sufficient before drawing conclusions about method superiority.

Estimating the number of components is a fundamental challenge in unsupervised learning, particularly when dealing with high-dimensional data with many components or severely imbalanced component sizes. This paper addresses this challenge for classical Gaussian mixture models. The proposed estimator is simple: center the data, compute the singular values of the centered matrix, and count those above a threshold. No iterative fitting, no likelihood calculation, and no prior knowledge of the number of components are required. We prove that, under a mild separation condition on the component centers, the estimator consistently recovers the true number of components. The result holds in high-dimensional settings where the dimension can be much larger than the sample size. It also holds when the number of components grows to the smaller of the dimension and the sample size, even under severe imbalance among component sizes. Computationally, the method is extremely fast: for example, it processes ten million samples in one hundred dimensions within one minute. Extensive experimental studies confirm its accuracy in challenging settings such as high dimensionality, many components, and severe class imbalance.

How much data is enough to make a scientific discovery? As biomedical datasets scale to millions of samples and AI models grow in capacity, progress increasingly depends on predicting when additional data will substantially improve performance. In practice, model development often relies on empirical scaling curves measured across architectures, modalities, and dataset sizes, with limited theoretical guidance on when performance should improve, saturate, or exhibit cross-over behavior. We propose a scaling-law framework for cross-modal discoverability based on spectral structure of data covariance operators, task-aligned signal projections, and learned representations. Many performance metrics, including AUC, can be expressed in terms of cumulative signal-to-noise energy accumulated across identifiable spectral modes of an encoder and cross-modal operator. Under mild assumptions, this accumulation follows a zeta-like scaling law governed by power-law decay of covariance spectra and aligned signal energy, leading naturally to the appearance of the Riemann zeta function. Representation learning methods such as sparse models, low-rank embeddings, and multimodal contrastive objectives improve sample efficiency by concentrating useful signal into earlier stable modes, effectively steepening spectral decay and shifting scaling curves. The framework predicts cross-over regimes in which simpler models perform best at small sample sizes, while higher-capacity or multimodal encoders outperform them once sufficient data stabilizes additional degrees of freedom. Applications include multimodal disease classification, imaging genetics, functional MRI, and topological data analysis. The resulting zeta law provides a principled way to anticipate when scaling data, improving representations, or adding modalities is most likely to accelerate discovery.

Factor-based Structural Equation Modeling (SEM) relies on likelihood-based estimation assuming a nonsingular sample covariance matrix, which breaks down in small-sample settings with $p>n$. To address this, we propose a novel estimation principle that reformulates the covariance structure into self-covariance and cross-covariance components. The resulting framework defines a likelihood-based feasible set combined with a relative error constraint, enabling stable estimation in small-sample settings where $p>n$ for sign and direction. Experiments on synthetic and real-world data show improved stability, particularly in recovering the sign and direction of structural parameters. These results extend covariance-based SEM to small-sample settings and provide practically useful directional information for decision-making.

Existing two-sample testing techniques, particularly those based on choosing a kernel for the Maximum Mean Discrepancy (MMD), often assume equal sample sizes from the two distributions. Applying these methods in practice can require discarding valuable data, unnecessarily reducing test power. We address this long-standing limitation by extending the theory of generalized U-statistics and applying it to the usual MMD estimator, resulting in new characterization of the asymptotic distributions of the MMD estimator with unequal sample sizes (particularly outside the proportional regimes required by previous partial results). This generalization also provides a new criterion for optimizing the power of an MMD test with unequal sample sizes. Our approach preserves all available data, enhancing test accuracy and applicability in realistic settings. Along the way, we give much cleaner characterizations of the variance of MMD estimators, revealing something that might be surprising to those in the area: while zero MMD implies a degenerate estimator, it is sometimes possible to have a degenerate estimator with nonzero MMD as well; we give a construction and a proof that it does not happen in common situations.

Many important tasks of large-scale recommender systems can be naturally cast as testing multiple linear forms for noisy matrix completion. These problems, however, present unique challenges because of the subtle bias-and-variance tradeoff of and an intricate dependence among the estimated entries induced by the low-rank structure. In this paper, we develop a general approach to overcome these difficulties by introducing new statistics for individual tests with sharp asymptotics both marginally and jointly, and utilizing them to control the false discovery rate (FDR) via a data splitting and symmetric aggregation scheme. We show that valid FDR control can be achieved with guaranteed power under nearly optimal sample size requirements using the proposed methodology. Extensive numerical simulations and real data examples are also presented to further illustrate its practical merits.