Reduces average-case complexity of sparse PCA from weak PC conjectures.
problem Characterizing the average-case complexity of sparse PCA.
method Reduction from planted clique conjecture to spiked covariance model.
result First full characterization of computational barrier in spiked covariance model, providing tight lower bounds at all sparsities.
Theoretical limits on verifying self-improving systems without risking unbounded utility.
problem Formalizing and proving the limits of safety verification for self-improving systems.
method Developed dual conditions and used Holder's inequality, NP counting method, and Lipschitz bounds to establish impossibility and ceiling results.
result A classifier-based safety gate cannot simultaneously permit unbounded beneficial self-modification and bounded cumulative risk.
Lower bounds show density estimation requires linear samples or query time.
problem Statistical-computational trade-offs in density estimation.
method Lower bound analysis on data structures.
result Lower bounds demonstrate statistical-computational trade-offs for density estimation.
Hardness result for approximating manifold radius.
problem Approximating the radius of triangulated manifolds.
method Proving NP-hardness for almost-polynomial approximation.
result It is NP-hard to approximate the hyperspherical radius up to an almost-polynomial factor.
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.
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.
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.
Generates hard graph instances for algorithm analysis and benchmarking.
problem Creating meaningful hard instances for algorithm analysis and benchmarking.
method HiSampler, a machine learning approach to learn hard instance distribution.
result HiSampler generates instances that are significantly harder than random-based approaches.
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
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.
Link crossing number problem is NP-hard.
problem Determining the crossing number of a link.
method NP-hardness proof.
result Link crossing number problem is NP-hard.
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.
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.
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.
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 …
Saccader improves hard attention models for vision tasks.
problem Challenges in training hard attention models with class label supervision.
method Proposes Saccader, a novel hard attention model trained with only class labels and policy gradient optimization.
result Achieves 75% top-1 and 91% top-5 accuracy while attending to less than one-third of the image.
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.
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.
Hard to estimate L2-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.
The Hard Lefschetz Theorem extends to certain Kähler Lie Algebroids with ellipticity.
problem Extending the Hard Lefschetz Theorem to Kähler Lie Algebroids.
method Analyzing a specific class of Kähler Lie Algebroids with ellipticity requirements.
result A class of Kähler Lie Algebroids satisfy the Hard Lefschetz Theorem with ellipticity.
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.
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.
We use Bayesian optimal experimental design to generate near-optimal attention sequences for faster hard attention training.
problem Training hard attention mechanisms is computationally expensive and high-variance.
method We frame hard attention as a BOED problem and use approximation methods from BOED to generate near-optimal attention sequences.
result Near-optimal attention sequences can speed up hard attention training and be reused by other networks.
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.
DC3 uses deep learning to solve hard-constrained optimization problems efficiently.
problem Hard constraints in optimization problems make classical solvers slow and infeasible.
method DC3 employs a differentiable procedure to enforce feasibility and unrolls corrections for inequality constraints.
result DC3 achieves near-optimal solutions while maintaining feasibility in both synthetic and real-world tasks.
Trading system uses NP-hard optimization to select stocks for high Sharpe ratio trading.
problem Finding profitable, uncorrelated stocks for high Sharpe ratio trading.
method NP-hard combinatorial optimization using Ising machine and simulated bifurcation algorithm.
result Trading strategy with FPGA-based system achieves 164 μs response latency.
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.
Study compact symplectic solvmanifolds' hard Lefschetz property.
problem Compact symplectic solvmanifolds' hard Lefschetz property.
method Analysis of compact symplectic solvmanifolds as quotients of solvable Lie groups by lattices.
result Characterization of conditions for the hard Lefschetz property.
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.
Hard thresholding remains efficient for DNN pruning, but smart pruning offers faster accuracy recovery.
problem Efficiently pruning deep neural networks while minimizing accuracy loss.
method Proposes a novel smart pruning algorithm based on difference of convex functions optimization.
result Smart pruning is often orders of magnitude faster than competing approaches while achieving low accuracy degradation.
AVH scores measure sample hardness, improving model calibration.
problem CNNs' poor calibration and overconfidence issues.
method AVH score based on normalized angular distance between feature embeddings and target classifiers.
result AVH scores correlate with human visual hardness and improve model calibration.
Study shows challenges in reinforcement learning math problems, proposing enhancements and a hardness measure.
problem Challenges in reinforcement learning finding rare high-reward instances.
method Combining combinatorial group theory, algorithmic enhancements, and topological hardness measure.
result Resolved mathematical questions and proposed enhancements for reinforcement learning.
Study shows computational hardness can improve adversarial robustness in learning.
problem Developing robust machine learning models against adversarial attacks.
method Investigate if computational limitations of attackers can enhance robustness.
result Demonstrated a learning task where computational robustness outperforms information-theoretic robustness.
Bailouts in financial networks are hard to optimize due to NP-hardness.
problem Optimizing bailouts in a network of insolvent banks.
method Modeling bailouts as an optimization problem, proving NP-hardness and inapproximability.
result Banks can strategically alter debt contracts to increase their market value in the event of a bailout.
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.
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.
Study on the topology of ordered disc configurations, revealing nontrivial homotopy classes.
problem Topology of ordered disc configurations and their homotopy types.
method Analysis of ordered configuration spaces of hard discs, focusing on homotopy types and nontrivial classes.
result Exhibit nontrivial classes in π_{n-3} for all n, and their persistence in deformed ambient discs.
The study constructs symplectic solvmanifolds satisfying the hard-Lefschetz condition.
problem Developing an analogue of Hodge theory for symplectic manifolds.
method Analyzing specific Lie algebras and their associated Lie groups, exploiting connections with Kneser graphs.
result Examples of almost-Kähler solvmanifolds satisfying the hard-Lefschetz condition are constructed.
We prove that for every d≥2, deciding if a pure, 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 and k≥0, deciding if a pure, d-dimensional, simplicial com…
Extends symplectic flow results to foliations.
problem Applying hard Lefschetz theorem to symplectic foliations.
method Generalizes results from symplectic flows to foliations.
result Extends transversal hard Lefschetz theorem to transversely symplectic foliations.
Paper improves deep learning for solving evolutionary equations with trainable hard constraints.
problem Low computational accuracy of standard PINNs in large temporal domains.
method Sequential learning strategies and trainable influence functions for hard constraints.
result Significantly improved computational accuracy and universality of the method.
There has recently been significant interest in hard attention models for tasks such as object recognition, visual captioning and speech recognition. Hard attention can offer benefits over soft attention such as decreased computational cost, but training hard attention models can be difficult because of the discrete la…
The paper defines and proves a new property for symplectic manifolds.
problem The study introduces a new property for symplectic manifolds.
method Defines and proves the L2-hard Lefschetz property for complete symplectic manifolds. result Proves that a complete symplectic manifold satisfies the L2-hard Lefschetz property if and only if every class of L2-harmonic forms contains a L2 symplectic harmonic form. Reviews g-theorem and hard Lefschetz theorem for face rings.
problem Understanding and proving the g-theorem and hard Lefschetz theorem for face rings.
method Perturbations of maps, biased Poincaré pairings, cobordism argument, edge-contractions.
result Alternative and alternative arguments for the Lefschetz property.
New symplectic structures found on complex manifolds without Kähler structures.
problem Finding symplectic structures on complex manifolds without Kähler structures.
method Constructing explicit lattices and cohomological computations.
result Compact complex manifolds with symplectic structures satisfying the Hard Lefschetz Condition.
We show that three natural decision problems about links and 3-manifolds are computationally hard, assuming some conjectures in complexity theory. The first problem is determining whether a link in the 3-sphere bounds a Seifert surface with Thurston norm at most a given integer; this is shown to be NP-complete. The sec…
The paper explores when linear system identification is hard or easy, especially for under-actuated systems.
problem Statistical hardness of learning linear systems, especially under-actuated or under-excited systems.
method Using tools from minimax theory and recent statistical tools for finite sample analysis of system identification.
result The controllability index of linear systems affects the sample complexity of identification, making some systems hard to learn.
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…