Derives an empirical capacity model for self-attention neural networks.
problem Theoretical capacity of large transformer models is not fully utilized by current optimization algorithms.
method Analyzes memory capacity of transformers using synthetic training data and common training algorithms.
result Derives an empirical capacity model (ECM) for a generic transformer.
Learning capacity measures model complexity, correlating with test loss and sample size.
problem Understanding model complexity and its relation to test performance.
method Formal correspondence between thermodynamics and inference; learning capacity as a measure of effective dimensionality.
result Learning capacity correlates with test loss and is a small fraction of model parameters.
Improves online learning algorithms for functional models with capacity assumptions.
problem Convergence rates of online stochastic gradient descent algorithms for functional linear models.
method Characterizations of slope function regularity, kernel space capacity, and sampling process covariance operator.
result Capacity assumptions can alleviate saturation of convergence rates as function regularity increases.
Introduces Rashomon Capacity to measure predictive multiplicity in probabilistic classifiers.
problem Predictive multiplicity in classification models leading to unjustified decisions.
method Introduces Rashomon Capacity, a metric for probabilistic classifiers, and provides a rigorous derivation.
result Rashomon Capacity captures nuanced score variations and provides strategies for disclosing conflicting models.
New analysis tightens memory capacity of Hopfield models using spherical codes.
problem Optimizing memory capacity in modern Hopfield models and Kernelized Hopfield Models.
method Connecting Hopfield models to spherical codes in information theory, establishing an optimal capacity bound and a sub-linear algorithm.
result First tight and optimal asymptotic memory capacity for modern Hopfield models, matching known lower bounds.
CapOptix uses options theory to price capacity in electricity markets.
problem Traditional capacity market designs fail to account for risk and price shocks.
method Interprets capacity commitments as reliability options and uses Markov Regime Switching Process.
result CapOptix provides more accurate pricing of capacity premia compared to existing mechanisms.
Memory capacity of DAM scales exponentially with feature separation, unaffected by correlations.
problem Understanding how feature correlations impact DAM's capacity.
method Developed an empirical framework to analyze DAM's capacity under varying feature correlations and pattern separations.
result Memory capacity scales exponentially with feature separation, unaffected by correlations.
Study proposes local effective dimension to measure model capacity and generalization error.
problem Capturing the generalization power of machine learning models.
method Proposes local effective dimension as a capacity measure.
result Local effective dimension bounds the generalization error and correlates well with it.
Study capacity constraints in continual learning with a simple model.
problem Understanding optimal resource allocation for agents with limited memory and compute resources.
method Analyzes a capacity-constrained linear-quadratic-Gaussian (LQG) sequential prediction problem and demonstrates optimal capacity allocation strategies.
result Derives a solution to the capacity-constrained LQG sequential prediction problem and shows how to optimally allocate capacity across sub-problems in the steady state.
High-capacity neural network ensembles often benefit more from high-capacity models than from increased diversity.
problem The performance of high-capacity neural network ensembles is often harmed by interventions that promote predictive diversity.
method A large-scale study of nearly 600 neural network classification ensembles, examining various interventions and architectures.
result Discouraging predictive diversity can be benign in large-network ensembles, and higher-capacity models often yield better performance than diverse architectures.
Capacity analysis has been recently introduced as a way to analyze how linear models distribute their modelling capacity across the input space. In this paper, we extend the notion of capacity allocation to the case of neural networks with non-linear layers. We show that under some hypotheses the problem is equivalent …
gLSTM improves graph neural networks by increasing storage capacity to prevent over-squashing.
problem Over-squashing in GNNs collapses information from a large receptive field into a single vector, creating an information bottleneck.
method Introduced a new synthetic task to measure over-squashing and adapted ideas from sequence modeling to develop gLSTM, a novel GNN architecture with improved capacity.
result gLSTM architecture demonstrates strong performance on synthetic and real-world graph benchmarks, mitigating over-squashing.
Managing data storage growth is of crucial importance to businesses. Poor practices can lead to large data and financial losses. Access to storage information along with timely action, or capacity forecasting, are essential to avoid these losses. In addition, ensuring high accuracy of capacity forecast estimates along …
The paper proposes a probabilistic autoencoder for discovering causal directions between variables.
problem Finding the causal direction between two associated variables.
method Building an autoencoder of the joint distribution and maximizing its estimation capacity relative to marginal distributions.
result The higher estimation capacity is consistent with the unconstrained choice of a distribution representing the cause, while the lower capacity reflects the constraints imposed by the mechanism on the distribution of the effect.
Remove symmetries to improve model optimization and performance.
problem Symmetries in loss functions trap models in low-capacity states, hindering training and optimization.
method Proposes syre, a simple algorithm to remove symmetries in neural networks.
result Removing symmetries correlates well with improved optimization and performance.
Study shows mass-capacity inequality for specific geometric manifolds.
problem Establishing mass-capacity inequality for certain geometric manifolds.
method Using conformally flat manifolds with nonnegative scalar curvature.
result Equality implies harmonically conformal to a specific subset of Euclidean space.
Study on neural networks' storage capacity and solution space structure.
problem Understanding the storage capacity and solution space structure of neural networks.
method Replica method from statistical physics.
result Storage capacity per parameter remains finite even with infinite width and weights exhibit negative correlations.
Model predicts COVID-19 growth in Senegal, highlighting health care capacity importance.
problem Impact of health care capacity on COVID-19 growth in Senegal.
method Compartmental model with logistic growth health care capacity, machine learning projection.
result Condition to avoid overwhelming health care system provided.
Modeling alignment as resource-limited cognitive processes, researchers derive performance bounds.
problem Systematic deviations in feedback-based alignment of large language models.
method Modeling alignment as a two-stage cascade UoHoY given S, with cognitive and total capacities. result Capacity-coupled Alignment Performance Interval derived from Fano and PAC-Bayes bounds.
Sharp estimates for p-capacity on manifolds with Ricci curvature bounds.
problem Estimating p-capacity on manifolds with Ricci curvature constraints.
method Sharp comparison inequalities, warped-product model ends, and scale-invariant quantities.
result Characterization of equality cases and optimal ranges for normalization parameters.
In this paper we address the following question, given a face representation, how many identities can it resolve? In other words, what is the capacity of the face representation? A scientific basis for estimating the capacity of a given face representation will not only benefit the evaluation and comparison of differen…
Recurrent neural networks are powerful models for processing sequential data, but they are generally plagued by vanishing and exploding gradient problems. Unitary recurrent neural networks (uRNNs), which use unitary recurrence matrices, have recently been proposed as a means to avoid these issues. However, in previous …
We study the volume of extrinsic balls and the capacity of extrinsic annuli in minimal submanifolds which are properly immersed with controlled radial sectional curvatures into an ambient manifold with a pole. The key results are concerned with the comparison of those volumes and capacities with the corresponding entit…
Designing neural network architectures is a task that lies somewhere between science and art. For a given task, some architectures are eventually preferred over others, based on a mix of intuition, experience, experimentation and luck. For many tasks, the final word is attributed to the loss function, while for some ot…
We study various capacities on compact Kähler manifolds which generalize the Bedford-Taylor Monge-Ampère capacity. We then use these capacities to study the existence and the regularity of solutions of complex Monge-Ampère equations.
Solves a discrete logarithmic Minkowski problem for electrostatic p-capacity.
problem Characterize measures generated by electrostatic p-capacity.
method Solves the discrete logarithmic Minkowski problem for 1 < p < n.
result Solves the discrete logarithmic Minkowski problem for measures in general position.
Enhances sequence memory capacity in neural networks.
problem Limited sequence capacity in Hopfield-like neural networks.
method Introducing a nonlinear interaction term and a generalized pseudoinverse rule.
result Significantly increased sequence capacity with novel scaling laws.
In this article, we propose the notion of the general p-affine capacity and prove some basic properties for the general p-affine capacity, such as affine invariance and monotonicity. The newly proposed general p-affine capacity is compared with several classical geometric quantities, e.g., the volume, the p-var…
While symplectic manifolds have no local invariants, they do admit many global numerical invariants. Prominent among them are the so-called symplectic capacities. Different capacities are defined in different ways, and so relations between capacities often lead to surprising relations between different aspects of sympl…
Study excess capacity in neural networks using Rademacher complexity.
problem Understanding how much capacity deep networks have beyond what's needed for classification.
method Unified Rademacher complexity bounds for function composition and convolutional layers, considering Lipschitz constants and initialization norms.
result There is substantial excess capacity per task, and capacity can be kept similar across different tasks.
Study rigidity by logarithmic capacity and related functions.
problem Rigidity phenomena in kernel functions and capacities.
method Exploration of Bergman kernel, logarithmic capacity, Green's function, and Euclidean distance/volume.
result Established rigidity theorems by logarithmic capacity.
Study binary perceptrons' capacity using random duality theory.
problem Characterize the capacity of binary perceptrons with general thresholds.
method Utilized fully lifted random duality theory (fl RDT) to characterize the capacity.
result Characterizations match replica symmetry breaking predictions and uncover the capacity for zero-threshold scenario.
New complete panel dataset for LMICs helps analyze innovation and development.
problem Lack of complete data for empirical analyses in LMICs.
method Predictive Mean Matching multiple imputation technique.
result Created a large dataset of 47 variables for 82 LMICs from 2005-2019.
Upper bounds for Lagrangian capacities of Liouville domains
problem Lagrangian capacity of Liouville domains
method Using S1-equivariant techniques result Extremal Lagrangian torus on the boundary of ellipsoid
New study shows how model complexity affects test risk, challenging classical theory.
problem Understanding how test risk scales with model complexity for large over-parametrized deep networks.
method Developed norm-based capacity measures for random features based estimators, providing precise characterization of estimator's norm concentration and test error.
result Predicted learning curve shows a phase transition from under- to over-parameterization, confirming classical U-shaped behavior with appropriate capacity measures.
Proves local maximizers for higher Ekeland-Hofer capacities in 4D star-shaped domains.
problem Finding local maximizers for higher Ekeland-Hofer capacities in specific domains.
method Analogous to 4D local Viterbo conjecture, proving maximizers for rational ellipsoids.
result Local maximizers of the k-th Ekeland-Hofer capacities are symplectomorphic to rational ellipsoids.
Develops a theory for mth order p-affine capacity for convex bodies containing the origin.
problem Defines and studies the mth order p-affine capacity for convex bodies containing the origin.
method Provides equivalent definitions, proves properties, and establishes inequalities.
result Establishes inequalities comparing to other geometric measures.
We introduce the concept of pseudo symplectic capacities which is a mild generalization of that of symplectic capacities. As a generalization of the Hofer-Zehnder capacity we construct a Hofer-Zehnder type pseudo symplectic capacity and estimate it in terms of Gromov-Witten invariants. The (pseudo) symplectic capacitie…
RAF model explains neural networks' dual rule learning and fact memorization.
problem Understanding how neural networks learn rules and memorize facts simultaneously.
method Introduces the Rules-and-Facts (RAF) model to bridge generalization and memorization.
result Characterizes conditions for simultaneous rule learning and fact memorization in neural networks.
Study relates symplectic homology capacity to periodic orbits in Liouville domains.
problem Relating symplectic homology capacity to periodic orbits in Liouville domains.
method Uses positive symplectic homology and Hofer-Zehnder capacity to establish bounds and existence of periodic points.
result Non-zero positive symplectic homology implies finite upper bound for Hofer-Zehnder capacity relative to skeleton and Hamiltonian diffeomorphisms.
New measure FTC quantifies how much a ReLU network can fine-tune.
problem Analyzing memorization capacity in fine-tuned neural networks.
method Defined Fine-Tuning Capacity (FTC) for additive fine-tuning of ReLU networks.
result Upper and lower bounds on FTC for 2 and 3-layer ReLU networks.
Study compares Monge-Ampère capacities on Kähler manifolds.
problem Comparing Monge-Ampère capacities on compact Kähler manifolds.
method Proved all capacities comparable, used Xia's integration by parts formula.
result All Monge-Ampère capacities are comparable.
Generalizes memory and forecasting capacities for nonlinear recurrent networks with dependent inputs.
problem Understanding memory and forecasting capabilities in networks with dependent inputs.
method Formulated bounds for memory and forecasting capacities in terms of network size and input properties.
result Proved that memory capacity for linear recurrent networks with independent inputs is given by the rank of the controllability matrix.
This paper is devoted to a geometric-measure-theoretic study of the brand new affine BV-capacity which is essentially different from the classic BV-capacity in dimension greater than one.
For any Lie group G, we construct a G-equivariant analogue of symplectic capacities and give examples when G=Tk×Rd−k, in which case the capacity is an invariant of integrable systems. Then we study the continuity of these capacities, using the natural topologies on the symplectic G-…
Study proves inequalities for mass-capacity on curved spaces.
problem Proving nonnegativity and positive lower bounds of mass on curved spaces.
method Applying mass-capacity inequalities from \cite{M22} to manifolds with nonnegative scalar curvature.
result Sufficient conditions for nonnegativity and positive lower bounds of mass.
Model for cross-border markets with limited transmission capacities.
problem Limited transmission capacities between two countries' markets.
method Developed a regime-switching process model with high-frequency approximation.
result Analytic tractability allows computation of key market quantities.
A new prior for VAEs improves model capacity by allowing a more flexible latent space.
problem Standard Gaussian priors in VAEs limit model capacity and performance.
method Proposed a Riemannian Brownian motion prior over a Riemannian structure of the latent space.
result The new prior significantly increases model capacity with only one additional scalar parameter.