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High-Dimensional Statistics
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High-Dimensional Statistics
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These lecture notes were written for the course 18.657, High Dimensional Statistics at MIT. They build on a set of notes that was prepared at Princeton University in 2013-14 that was modified (and hopefully improved) over the years.
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Cited by 18 Pith papers
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Resolution of the Detection Threshold Conjecture for Random Geometric Graphs in the $d>n$ Regime
Proves detection of RGG vs. ER is impossible for d ≫ (n h(p))^3 and d ≥ (1+ε)n, resolving the detection threshold conjecture in the regime p ≳ n^{-2/3}/log n.
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Detection Is Harder Than Estimation in Certain Regimes: Inference for Moment and Cumulant Tensors
The minimax rate for estimating d-th order moment tensors is sqrt(p/n) wedge 1, while low-degree evidence shows detection of vanishing cumulants is hard for n much less than p to the d/2, creating a reverse detection-...
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What is the long-run distribution of stochastic gradient descent? A large deviations analysis
SGD's stationary distribution is Boltzmann-Gibbs with temperature equal to step-size, concentrating exponentially on minimum-energy critical points.
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Variance-Reduced Q-Learning over Static and Time-Varying Networks
VRDQ achieves the optimal collaborative error rate 1/√(NT) for decentralized tabular Q-learning while requiring only O(log²(NT)) communication per agent, on both static and time-varying networks.
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Tight $L_\infty$ Sample Complexity for Low-Degree and Sparse Boolean Polynomials
Minimax sample complexity for uniform L_infty estimation is Theta(n^{d+1}) for degree-d polynomials and Theta(ns^2) for s-sparse Fourier-Walsh polynomials under noise, exceeding noiseless rates by factors of n and s.
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The End Justifies the Mean: A Linear Ranking Rule for Proportional Sequential Decisions
The angular mean of voter scoring vectors satisfies long-run individual proportionality for sequential linear ranking decisions.
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Robust Uniform Recovery of Structured Signals from Nonlinear Observations
RAIC unifies uniform recovery of structured signals from nonlinear observations via PGD, yielding error rates comparable to nonuniform guarantees up to log factors in sparse and 1-bit settings.
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Spectral recovery of a planted triangle-dense subgraph
Spectral and SDP algorithms using signed triangle count matrices recover planted triangle-dense subgraphs, with computational threshold at least sqrt(n) for low-degree methods versus information-theoretic threshold at...
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FLAG: Foundation model representation with Latent diffusion Alignment via Graph for spatial gene expression prediction
FLAG redefines spatial gene expression prediction as structured distribution modeling via latent diffusion, integrating a spatial graph encoder and GFM alignment to capture gene-gene and spatial relationships, with ne...
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Calibrated Estimation and Inference for Semiparametric Regression Models
SABRE calibrates a B-spline initial estimator against its simulated expectation, reducing finite-sample bias in semiparametric regression without inflating variance.
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Optimal Online and Offline Algorithms for Contextual MNL with Applications to Assortment and Pricing
New algorithms for joint contextual MNL assortment and pricing deliver improved online regret bounds of order W sqrt(d T log N)/L0 and local suboptimality guarantees offline.
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Robust mean estimation under star-shaped constraints with heavy-tailed noise
Minimax squared l2 error for robust mean estimation under star-shaped sets with heavy-tailed noise and contamination level ε is max(δ*², ε σ²) ∧ d², where δ* is the largest scale satisfying N δ²/σ² ≤ log M^loc(δ, c).
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How Does the Lagrangian Guide Safe Reinforcement Learning through Diffusion Models?
ALGD augments the Lagrangian to locally convexify the energy landscape in diffusion models, stabilizing safe RL training and generation without changing optimal policies.
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Newsvendor under Ambiguity and Misspecification
Derives closed-form optimal order quantity for newsvendor under mean-variance ambiguity and optimal-transport misspecification, generalizing Scarf model with finite-sample performance guarantees separating estimation ...
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LoRA vs. Full Fine-Tuning: A Theoretical Perspective
In linear regression, LoRA can achieve lower excess risk than full fine-tuning when the pretraining-downstream difference is low-rank, and small LoRA ranks can improve generalization by acting as regularization.
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Existence of Invariant Probability Measures for Stochastic Differential Equations with Finite Time Delay
Sufficient conditions are given for invariant probability measures in stochastic delay differential equations driven by integrable Lévy noise, with a key reduction from bounded solutions to bounded segments under a on...
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Stochastic Mackey-Glass Equations and Other Negative Feedback Systems: Existence of Invariant Measures
Non-trivial invariant measures exist for stochastic Mackey-Glass and Nicholson's blowflies equations if and only if solutions remain bounded away from zero in probability for at least one initial condition.
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Calibrated Estimation and Inference for Semiparametric Regression Models
SABRE is a simulation-based bias correction framework that reduces finite-sample bias for the parametric component and dispersion parameter in semiparametric regression models, with asymptotic bias reduction without v...
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