Low-degree method fails to predict robust subspace recovery problem.
problem Predicting computational tractability of robust subspace recovery problem.
method Low-degree polynomial framework, anti-concentration properties.
result Low-degree method fails to predict computational tractability of robust subspace recovery problem even up to high degree.
Sublinear algorithms detect cliques in graphs with high probability.
problem Detecting a planted clique in random graphs efficiently.
method Non-adaptive low-degree polynomial queries of adjacency matrix entries.
result Sublinear time detection is possible for a specific range of clique sizes.
Study disproves conjecture about low-degree polynomials in hypothesis testing.
problem Conjecture about limitations of polynomial-time algorithms in hypothesis testing.
method Used counterexamples to refute the conjecture and modified the conjecture to rule out the counterexample.
result Disproved conjecture about limitations of low-degree polynomials in hypothesis testing.
Statistical query algorithms and low-degree tests are nearly equivalent in high-dimensional hypothesis testing.
problem High-dimensional hypothesis testing and information-computation gaps.
method Analysis of statistical query framework and low-degree polynomials.
result Statistical query algorithms and low-degree polynomials are almost equivalent in power under mild conditions.
Paper connects free-energy and low-degree hardness in high-dimensional statistics.
problem High-dimensional statistical inference problems are computationally hard.
method Defines a free-energy criterion and connects it to low-degree hardness.
result Establishes connection between free-energy and low-degree hardness for Gaussian models.
New findings on computational limits for estimating hidden structures.
problem Estimating hidden structures in noisy data.
method Use of low-degree polynomials as a restricted model of computation.
result Established low-degree hardness of recovery problems for easy detection problems.
New work shows FP potential monotonicity equals low-degree polynomial estimators limits.
problem Establishing a precise mathematical relationship between statistical physics and polynomial estimators limits.
method Analyzing Gaussian additive models (GAMs) to show FP potential monotonicity equals low-degree polynomial estimators limits.
result For a broad family of Gaussian additive models, the power of low-degree polynomials is equivalent to the monotonicity of the annealed FP potential.
New method explains computational barriers in high-dimensional statistical models.
problem Understanding detection-recovery gaps in high-dimensional inference.
method Combining algorithmic contiguity and cross-validation reduction to obtain conditional computational lower bounds.
result Mild control of low-degree advantage is sufficient to explain computational barriers for recovery.
New algorithms find half-optimal independent sets in sparse graphs.
problem Finding large independent sets in sparse random graphs.
method Low-degree polynomial algorithms.
result Low-degree polynomial algorithms can find independent sets of half-optimal size.
Paper proves computational hardness for graph matching and detection problems.
problem Computational hardness for graph matching and detection problems in correlated random graphs.
method Algorithmic contiguity and low-degree advantage bounds.
result No efficient algorithms exist for certain graph matching and detection problems.
New study shows low-degree polynomial algorithms struggle at clause densities close to Fix's.
problem Finding satisfying assignments in random k-SAT formulas at high clause densities.
method Analysis of low-degree polynomial algorithms and a new many-way overlap gap property.
result No efficient algorithms can find satisfying assignments at clause densities close to Fix's.
These notes survey and explore an emerging method, which we call the low-degree method, for predicting and understanding statistical-versus-computational tradeoffs in high-dimensional inference problems. In short, the method posits that a certain quantity -- the second moment of the low-degree likelihood ratio -- gives…
Detection of dense cycles in graphs reveals a gap between easy detection and hard recovery.
problem Detecting and recovering dense cycles in Erdős-Rényi graphs.
method Characterization of computational thresholds for detection and recovery using low-degree polynomial algorithms.
result A gap exists between the detection and recovery thresholds for certain parameter regimes.
Study shows a tradeoff between sample complexity and computational efficiency for learning halfspaces with random noise.
problem PAC learning γ-margin halfspaces with Random Classification Noise.
method Established an information-computation tradeoff and provided a simple efficient algorithm with sample complexity O(1/(γ^2 ε^2)). Also, proved lower bounds for SQ algorithms and low-degree polynomial tests.
result Inherent gap between sample complexity and computational efficiency for learning halfspaces with random noise.
Survey on using low-degree polynomials to assess statistical tasks complexity.
problem Understanding the complexity of statistical tasks using polynomial functions.
method Applying low-degree polynomials to measure the complexity of statistical tasks, including detection, recovery, and estimation.
result Low-degree polynomials provide a framework to predict and explain statistical-computational tradeoffs.
New method uses almost orthonormal bases to prove low-degree lower bounds in complex statistical models.
problem Proving statistical-computational gaps in high-dimensional models with planted structures.
method Constructing an almost orthonormal polynomial basis under the planted distribution.
result Established new low-degree lower bounds for various complex models.
The paper explores how low-degree polynomials can detect shuffled linear regression models.
problem Detecting multivariate shuffled linear regression models from independent Gaussian random matrices.
method Investigates the effectiveness of low-degree polynomial algorithms for distinguishing the model from independent Gaussian random matrices.
result Establishes a phase transition phenomenon in the performance of low-degree polynomial algorithms for distinguishing the model.
New study shows limits of low-degree algorithms in finding large independent sets in sparse hypergraphs.
problem Finding large independent sets in sparse random hypergraphs.
method Low-degree polynomial algorithms are analyzed to determine their limits.
result Low-degree algorithms can find independent sets of density up to \(\left(\frac{\log d}{(r-1)d}
ight)^{1/(r-1)}\), but no larger.
Learn low-degree functions with few random queries.
problem Learning low-degree functions from limited random queries.
method Learn bounded functions f : { − 1 , 1 } n o [ − 1 , 1 ] f:\{-1,1\}^n o[-1,1] f : { − 1 , 1 } n o [ − 1 , 1 ] of degree at most d d d with L 2 L_2 L 2 -accuracy ε \varepsilon ε and confidence 1 − δ 1-δ 1 − δ from log ( f r a c n δ ) ε − d − 1 C d 3 / 2 log d \log( frac{n}δ)\,\varepsilon^{-d-1} C^{d^{3/2}\sqrt{\log d}} log ( f r a c n δ ) ε − d − 1 C d 3/2 l o g d random queries. result Learn low-degree functions efficiently with logarithmic number of random queries.
New algorithm achieves optimal clustering for sparse centers with high dimensions.
problem Statistical and computational limits of clustering sparse centers with high dimensions.
method Sparse clustering algorithm based on sparse PCA.
result Achieves minimax optimal misclustering rate under certain conditions.
New computational lower bounds for clustering and related problems.
problem Statistical-computational gaps in high-dimensional clustering problems.
method Investigation of low-degree polynomials in latent space models to derive lower bounds.
result New and sharper computational lower bounds for clustering, sparse clustering, and biclustering.
New lower bounds show challenges in clustering in moderate dimensions.
problem Clustering points from mixtures of isotropic Gaussians in moderate dimensions.
method Established low-degree polynomial lower bounds and developed a novel non-spectral algorithm.
result New lower bounds reveal a 'non-parametric rate' in moderate dimensions.
Gradient Descent with Projection learns low-degree polynomials efficiently.
problem Learning low-degree spherical polynomials with neural networks.
method Over-parameterized two-layer neural network with Gradient Descent with Projection.
result Achieves nearly minimax optimal sample complexity and risk bound.
New evidence shows computational barriers in graphon estimation using low-degree polynomials.
problem Estimating graphons efficiently and accurately.
method Low-degree polynomials to analyze computational limits.
result Low-degree polynomial estimators cannot significantly outperform USVT in graphon estimation.
Deep learning explained through spectral filtering of hierarchical features.
problem Understanding how deep neural networks learn useful representations from data.
method Neural Low-Degree Filtering (Neural LoFi) as a stylized limit of gradient-based training.
result Predicts how representations are selected layer by layer and explains emergence of concepts.
Detects dense subhypergraphs in random hypergraphs using low-degree polynomials.
problem Detecting a planted dense subhypergraph in a random hypergraph model.
method Degree-n^o(1) polynomials of adjacency tensor entries.
result Thresholds for detection in different density regimes.
We introduce the problem of learning mixtures of k k k subcubes over { 0 , 1 } n \{0,1\}^n { 0 , 1 } n , which contains many classic learning theory problems as a special case (and is itself a special case of others). We give a surprising n O ( log k ) n^{O(\log k)} n O ( l o g k ) -time learning algorithm based on higher-order multilinear moments. It is not possible to l…
Two-layer NN with channel attention learns low-degree spherical polynomials efficiently.
problem Learning low-degree spherical polynomials with over-parameterized neural networks.
method Two-layer neural network with channel attention, vanilla gradient descent, learnable channel selection.
result Minimally improved sample complexity of $n \asymp Θ(d^{\ell_0}/\eps)$ for learning low-degree polynomials.
Paper proves first non-trivial PTF testing lower bounds for NGCA.
problem Proving lower bounds against PTF tests is challenging.
method Developed tools to prove PTF testing lower bounds for NGCA.
result First non-trivial PTF testing lower bounds for NGCA.
New findings show that common optimization algorithms struggle with random problems.
problem Finding near-optimal solutions to random optimization problems.
method Low-degree polynomials, Boolean circuits, and Langevin dynamics.
result These algorithms fail to produce nearly optimal solutions with high probability.
We propose an efficient meta-algorithm for Bayesian estimation problems that is based on low-degree polynomials, semidefinite programming, and tensor decomposition. The algorithm is inspired by recent lower bound constructions for sum-of-squares and related to the method of moments. Our focus is on sample complexity bo…
GCNs favor high-degree nodes, leading to biased performance; a new method mitigates this.
problem Degree-related biases in GCNs, especially for low-degree nodes.
method Developed a novel SL-DSGC that reduces model and data biases.
result SL-DSGC improves GCN accuracy significantly for low-degree nodes.
Unified approach to tensor PCA and related problems using tensor cumulants.
problem Statistical inference on invariant distributions, particularly tensor PCA.
method Definition and analysis of tensor cumulants to unify and extend previous results.
result Unified explanation of hardness and subexponential-time algorithms for tensor PCA.
New findings on tensor decomposition complexity, showing polynomial functions can estimate the largest component under certain conditions.
problem The complexity of tensor decomposition, especially for low-degree polynomials.
method Modeling a slightly larger component in a random tensor decomposition and using polynomial functions to estimate it.
result Polynomial functions can accurately estimate the largest component when r ≪ n 3 / 2 r \ll n^{3/2} r ≪ n 3/2 but fail when r ≫ n 3 / 2 r \gg n^{3/2} r ≫ n 3/2 . New algorithm learns PTFs with noisy data efficiently.
problem Learning low-degree PTFs with noisy data efficiently.
method Structural result and novel robust Chow vector estimation.
result PAC learns PTFs with nasty noise using efficient samples.
Study on estimating Gaussian mean with missing data in high dimensions.
problem Estimating Gaussian mean in high dimensions with missing data due to realizable contamination.
method Statistical Query model, Low-Degree Polynomials, PTF tests, and algorithms.
result Established information-computation gap and developed efficient algorithms.
GANs learn distributions by matching low-degree moments.
problem Understanding when GANs learn the target distribution efficiently.
method Theoretical analysis and empirical observation of GAN training process.
result GANs can learn notable distributions by matching polynomially many low-degree moments.
Study shows computational and statistical gaps in Gaussian Single-Index Models.
problem Statistical and computational trade-offs in high-dimensional regression problems.
method Analysis of SQ and LDP frameworks, partial-trace algorithm.
result Computational algorithms require significantly more samples than information-theoretic limits.
Classifies real rational knots and curves in a specific quadric space.
problem Classifying real rational knots and curves in a quadric space of signature ( 3 , 2 ) (3,2) ( 3 , 2 ) . method Classification through a study of real rational curves of low degree in the quadric.
result Provides representatives of all real rational knots of degree ≤ 5 \leq 5 ≤ 5 in the quadric. New algorithms learn multi-index models via harmonic analysis, achieving statistical and computational trade-offs.
problem Learning multi-index models with unknown projections of input data.
method Exploiting the equivariance of the problem under the orthogonal group, we derive lower bounds and construct spectral algorithms based on harmonic tensor unfolding.
result Achieve statistical and computational trade-offs between sample and runtime complexity.
The resilience of low-degree Rademacher chaos is studied, providing probabilistic lower bounds.
problem Understanding how much a Rademacher chaos can withstand adversarial sign-flips without significant probability changes.
method Probabilistic lower-bound guarantees for the resilience of Rademacher chaos of arbitrary degree.
result Probabilistic lower-bound guarantees for the resilience of Rademacher chaos of arbitrary degree, especially meaningful for constant degree.
The paper finds new graph covers with exceptionally low degree.
problem Graph covers with primitive homology not spanned by lifts.
method Focused on finite p p p -group deck groups, developed a character table-based algorithm. result Found the smallest known nontrivial examples with covering degree 128.
Polynomial-time algorithm estimates mean with bounded covariance using differential privacy.
problem Estimating mean of a d-variate distribution with differential privacy constraints.
method Sum of Squares (SoS) exponential mechanism for polynomial-time differentially private estimation.
result First polynomial-time algorithm with O ( d ) O(d) O ( d ) samples for mean estimation under pure differential privacy. New algorithm reduces contamination in supervised learning.
problem Learning with contamination in supervised learning.
method Iterative polynomial filtering.
result Efficient learning of functions with contamination.
We construct families of hyperbolic hypersurfaces X d ⊂ P n + 1 ( C ) X_d\subset\mathbb{P}^{n+1}(\mathbb{C}) X d ⊂ P n + 1 ( C ) of degree d ≥ ( n + 3 2 ) 2 d\geq {\textstyle{(\frac{n+3}{2})^2}} d ≥ ( 2 n + 3 ) 2 .
New method for estimating sparse means in noisy data.
problem Estimating the mean of a sparse distribution in the presence of outliers.
method Difference-of-Pairs Filtering technique for list-decodable sparse mean estimation.
result First sample and computationally efficient algorithm for list-decodable sparse mean estimation.
Study efficient estimation of hidden subspaces in Gaussian Multi-index models.
problem Estimating hidden subspaces in Gaussian Multi-index models with low-dimensional projections.
method Introduced the generative leap exponent and developed an agnostic sequential estimation procedure using spectral U-statistics.
result Achieved optimal sample complexity of $n=Θ(d^{1 \vee \k/2})$ for efficient estimation.
We prove Minding's Theorem for C 2 C^2 C 2 -immersions with constant negative Gauss curvature. As a Corollary we also prove Minding's Theorem for C 1 M C^{1M} C 1 M -immersions in the sense of \cite{DS}.