Paper proves hardness of learning various complex models under local pseudorandom generators.
problem Hardness of learning various complex models.
method Existence of local pseudorandom generators.
result Proves hardness of learning shallow ReLU neural networks and other models.
Moving between 3-manifold triangulations is NP-hard
problem Moving between two triangulations of a 3-manifold
method Showing that the number of bistellar moves and sparse degree-two edge collapses is NP-hard
result First NP-hardness result concerning moves between two triangulations of a 3-manifold
This work connects hardness of approximation and learning.
problem Hardness of approximation and learnability in machine learning.
method Shows a single hardness property implying both approximation and learning hardness.
result Obtains new results on hardness of approximation and learnability of specific functions.
Hardness proven for learning neural networks with polynomial size and Gaussian inputs.
problem Learning one hidden layer ReLU neural networks with polynomial size and Gaussian inputs.
method Based on the hardness of the Continuous Learning with Errors (CLWE) problem.
result Hardness of learning neural networks is proven under standard cryptographic assumptions.
The paper proves shellability is hard for d-balls when d is at least 3.
problem Shellability for d-balls is NP-hard when d ≥ 3.
method NP-hardness proof for triangulated d-balls and d-manifolds/d-pseudomanifolds with boundary.
result Shellability is NP-hard for triangulated d-balls when d ≥ 3.
Hardness proven for neural networks with natural weights.
problem Difficulty in learning neural networks with weights from natural distributions.
method Proved hardness for depth-2 networks with natural weights distributions.
result Most networks are hard to learn with natural weights.
Hard to estimate L 2 L^2 L 2 -accurate scores without strong assumptions.
problem Estimating the score of unknown data distributions accurately.
method Reduction to generating samples and leveraging lattice-based cryptography hardness.
result Score estimation is computationally hard even with polynomial sample complexity.
In this thesis I explore challenging discrete energy minimization problems that arise mainly in the context of computer vision tasks. This work motivates the use of such "hard-to-optimize" non-submodular functionals, and proposes methods and algorithms to cope with the NP-hardness of their optimization. Consequently, t…
Researchers prove NP-hardness of learning parameter-bounded Bayes nets.
problem Learning parameter-bounded Bayes nets is computationally hard.
method Proved NP-hardness of learning parameter-bounded Bayes nets and a promise search variant.
result Proved NP-hardness of a promise search variant of LEARN.
Meta-learning strategy improves few-shot classification performance.
problem Few-shot classification with deep neural networks struggles when labeled samples are limited.
method Proposes an easy-to-hard expert meta-training strategy to arrange training tasks based on task hardness.
result Meta-learners achieve better results with the proposed expert training strategy.
We show that it is N P \mathsf{NP} NP -hard to approximate the hyperspherical radius of a triangulated manifold up to an almost-polynomial factor.
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.
HardCoRe-NAS finds fitting neural networks adhering to hard resource constraints.
problem Finding fitting neural networks that adhere to hard resource constraints.
method Accurate formulation of resource requirement and scalable search method.
result HardCoRe-NAS generates state-of-the-art architectures strictly satisfying hard resource constraints.
The paper proves a generalized Lefschetz duality for a specific type of manifold.
problem Proving the hard Lefschetz duality for a new class of manifolds.
method Generalizing Kähler identities to prove the duality for locally conformally almost Kähler manifolds.
result The hard Lefschetz duality is established for locally conformally almost Kähler manifolds.
Study shows exponential sample complexity for stabilizing certain linear systems.
problem Statistical hardness of learning to stabilize linear time-invariant systems.
method Analysis of sample complexity and co-stabilizability using robust control ideas.
result Sample complexity increases exponentially with system dimension.
Paper investigates hardness of learning neural networks under manifold hypothesis.
problem Hardness of learning neural networks under the manifold hypothesis.
method Extending proofs of hardness in the SQ and cryptographic settings to the geometric setting.
result Learning is hard under input manifolds of bounded curvature but learnable with additional assumptions on manifold volume.
New unsupervised method selects hard negative samples for contrastive learning.
problem How to select good negative examples for contrastive learning without using true similarity information.
method Developed a new family of unsupervised sampling methods for hard negative selection.
result Improves downstream performance across multiple modalities.
Detecting adversarial examples is as hard as classifying them.
problem The difficulty of detecting adversarial examples in machine learning models.
method Proved a general hardness reduction between detection and classification of adversarial examples.
result The hardness reduction implies that detecting adversarial examples is computationally infeasible.
Study on hard Legendrian unknots using normal rulings.
problem Understanding the complexity of Legendrian unknots in knot theory.
method Using normal rulings to obstruct and construct hard unknot diagrams.
result Construction of infinitely many smoothly hard max-tb unknot diagrams with bounds on minimum possible writhe.
Improved hardness results for clearing payments in financial networks with CDSs.
problem Determining clearing payments in financial networks with CDSs after financial shocks.
method Analyzing computational complexity of clearing problems, showing PPAD-hardness and FIXP-completeness improvements.
result PPAD-hardness of clearing problem significantly improved to ε ≈ 0.101.
We prove that for every d ≥ 2 d\geq 2 d ≥ 2 , deciding if a pure, d d d -dimensional, simplicial complex is shellable is NP-hard, hence NP-complete. This resolves a question raised, e.g., by Danaraj and Klee in 1978. Our reduction also yields that for every d ≥ 2 d \ge 2 d ≥ 2 and k ≥ 0 k \ge 0 k ≥ 0 , deciding if a pure, d d d -dimensional, simplicial com…
New findings show learning deeper neural networks is hard even with Gaussian inputs and non-degenerate weights.
problem The computational complexity of learning neural networks, especially deeper ones.
method Smoothed analysis framework and local pseudorandom generators.
result Learning depth-3 ReLU networks under Gaussian input distribution is hard even if weight matrices are non-degenerate.
Hard instances, which require a long time for a specific algorithm to solve, help (1) analyze the algorithm for accelerating it and (2) build a good benchmark for evaluating the performance of algorithms. There exist several efforts for automatic generation of hard instances. For example, evolutionary algorithms have b…
Tackles the computational hardness of HPC detection, conjecturing equivalence to PC detection.
problem Computational hardness of hypergraphic planted clique detection.
method No specific method mentioned; focuses on conjecturing equivalence.
result Equivalence of computational hardness between HPC and PC detection.
Hardness proof for agnostically learning halfspaces from worst-case lattice problems.
problem Agnostically learning halfspaces in the presence of noise.
method Reduction to worst-case lattice problems (GapSVP, SIVP).
result No efficient algorithm can achieve misclassification error better than 1/2 - γ under given hardness assumptions.
Three hard diagrams of the unknot require extra crossings to simplify.
problem Finding diagrams of the unknot that require many crossings to simplify.
method Applying previously proposed methods to construct diagrams and using computational resources to prove their hardness.
result Three hard diagrams of the unknot require at least three extra crossings.
In this article supervised learning problems are solved using soft rule ensembles. We first review the importance sampling learning ensembles (ISLE) approach that is useful for generating hard rules. The soft rules are then obtained with logistic regression from the corresponding hard rules. In order to deal with the p…
The use of M-estimators in generalized linear regression models in high dimensional settings requires risk minimization with hard L 0 L_0 L 0 constraints. Of the known methods, the class of projected gradient descent (also known as iterative hard thresholding (IHT)) methods is known to offer the fastest and most scalable sol…
Developed a new thresholding method that connects soft and hard thresholding.
problem Connecting soft and hard thresholding methods in data analysis.
method Scaled soft thresholding method with empirical scaling values.
result Found two sources of over-fitting in the scaled soft thresholding method.
For a Lie group G = R n ⋉ φ R m G=\R^{n}\ltimes_φ\R^{m} G = R n ⋉ φ R m with the semi-simple action φ : R n → A u t ( R m ) φ:\R^{n}\to {\rm Aut}(\R^{m}) φ : R n → Aut ( R m ) , we show that if Γ Γ Γ is a finite extension of a lattice of G G G then K ( Γ , 1 ) K(Γ, 1) K ( Γ , 1 ) is formal. Moreover we show that a compact symplectic aspherical manifold with the fundamental group Γ Γ Γ satisfies the hard Lefschetz proper…
Study categorizes knots and links as rigid or shaky based on Reidemeister moves.
problem Classifying knots and links as rigid or shaky based on adaptability to Reidemeister moves.
method Categorization of hard diagrams as rigid or shaky, investigation of rigid and shaky hard diagrams for specific knots and links.
result Every link has a rigid hard diagram, and there is an upper limit for the number of crossings in such diagrams.
Proves hard Lefschetz theorem and Hodge-Riemann relations for convex valuations.
problem Proving properties of convex valuations analogous to Kähler manifolds.
method Elliptic operator theory and perturbation theory applied to unbounded operators on a Hilbert space.
result Establishes hard Lefschetz theorem and Hodge-Riemann relations for convex bodies.
This paper shows how optimizing with hard negative examples improves image retrieval.
problem Training with hard negative examples leads to poor training behavior.
method Characterize the space of triplets, derive why hard negatives fail, and offer a fix to the loss function.
result Optimizing with hard negative examples leads to more generalizable features and better image retrieval.
We describe a method for generating minimal hard prime surface-link diagrams. We extend the known examples of minimal hard prime classical unknot and unlink diagrams up to three components and generate figures of all minimal hard prime surface-unknot and surface-unlink diagrams with prime base surface components up to …
An algorithmically hard phase was described in a range of inference problems: even if the signal can be reconstructed with a small error from an information theoretic point of view, known algorithms fail unless the noise-to-signal ratio is sufficiently small. This hard phase is typically understood as a metastable bran…
Paper characterizes causal graphs from hard interventions and proposes a learning algorithm.
problem Discovering causal structure from hard interventions and observational data.
method Proposes graphical constraints and a learning algorithm based on do-calculus.
result Characterizes interventional equivalence classes of causal graphs with latent variables.
We study the transversal hard Lefschetz theorem on a transversely symplectic foliation. This article extends the results of transversally symplectic flows (H.K.~Pak, "Transversal harmonic theory for transversally symplectic flows", J. Aust. Math. Soc. 84 (2008), 233--245) to the general transversely symplectic foliatio…
This work tackles learning Markov games with adversarial opponents and achieves both average reward and exploitation.
problem Achieving both average reward and exploiting adaptive opponents in Markov games.
method Develops efficient algorithms and proves hardness results for learning Markov games with adversarial opponents.
result Achieves K \sqrt{K} K -regret bounds for certain conditions on opponent policies, complemented by an exponential lower bound. New findings show some symplectic solvmanifolds fail hard-Lefschetz condition.
problem Characterizing symplectic solvmanifolds that do not satisfy the hard-Lefschetz condition.
method Detailed analysis of Lie algebra cohomology groups and construction of lattices.
result Symplectic solvmanifolds with non-semisimple actions fail the hard-Lefschetz condition at degree 1 or 2.
Iterative hard thresholding (IHT) is a projected gradient descent algorithm, known to achieve state of the art performance for a wide range of structured estimation problems, such as sparse inference. In this work, we consider IHT as a solution to the problem of learning sparse discrete distributions. We study the hard…
Study on learning halfspaces under adversarial perturbations, finding computational hardness.
problem Learning halfspaces in the presence of adversarial noise.
method Introduced an efficient learning algorithm and proved a nearly matching computational hardness result.
result The L ∞ L_{\infty} L ∞ perturbations case is provably computationally harder than 2 ≤ p < ∞ 2 \leq p < \infty 2 ≤ p < ∞ . Training neural networks is hard in fixed dimensions.
problem Training two-layer neural networks is computationally hard in fixed dimensions.
method Parameterized complexity analysis considering dimension and number of neurons.
result Training two-layer neural networks is NP-hard for two dimensions.
Computing PL geometric category in 2D is NP-hard.
problem Determining the PL geometric category of 2D polyhedra.
method Reduction from shellability of 2-complexes, which is known to be NP-hard.
result It is NP-hard to decide whether the PL geometric category of a 2D polyhedron is at most 2.
MAML outperforms NAL in diverse task landscapes.
problem Understanding when and how MAML outperforms NAL in various task landscapes.
method Analytical and numerical studies in a linear regression setting with a mixture of easy and hard tasks.
result MAML gains over NAL when there is task hardness discrepancy and optimal solutions of hard tasks are closely packed.
Reduces learning periodic neural networks to lattice problems, proving hardness under cryptographic assumptions.
problem Learning single periodic neurons in noisy environments.
method Reduction to worst-case lattice problems, using LLL algorithm.
result Polynomial-time algorithms for learning these functions are hard under cryptographic assumptions.
We study the properties of the multiplicative structure on valuations on convex sets. We prove a new version of the hard Lefschetz theorem for even translation invariant continuous valuations, and discuss related problems of integral geometry. Then we formulate a conjectural analogue of this result for odd valuations.
We show that {\sc Heegaard Genus ≤ g \leq g ≤ g }, the problem of deciding whether a triangulated 3-manifold admits a Heegaard splitting of genus less than or equal to g g g , is NP-hard. The result follows from a quadratic time reduction of the NP-complete problem {\sc CNF-SAT} to {\sc Heegaard Genus ≤ g \leq g ≤ g }.
Introduces a continuous version of LWE problem.
problem Hardness of learning mixtures of Gaussians.
method Polynomial-time quantum reduction from CLWE to lattice problems.
result CLWE shares hardness with LWE.