The study calculates the injectivity capacity of ReLU networks using a novel mathematical approach.
problem Determining the injectivity capacity of ReLU networks layers.
method Employing fully lifted random duality theory (fl RDT) to handle the ℓ0 spherical perceptron and implicitly the ReLU layers injectivity. result The lifting mechanism converges remarkably fast with relative corrections not exceeding 0.1%.
Deep ReLU networks show that 4 layers suffice for unique input recovery.
problem Injectivity capacity of deep ReLU networks.
method Developed a program connecting deep ReLU injectivity to an l-extension of the ℓ0 spherical perceptrons, using random duality theory. result Only 4 layers are needed for unique input recovery, showing expansion saturation effect.
Hybrid framework injects TSLM insights into GRLM for robust time-series reasoning.
problem Lack of domain-specific knowledge in large language models for time-series reasoning.
method Hybrid knowledge-injection framework combining RLVR for efficient knowledge transfer.
result Consistently outperforms existing models by 7.9%-26.1% on multivariate time-series benchmarks.
Flexible VAEs using FIFs improve model likelihood on image datasets.
problem Limitations of diagonal Gaussian posteriors in VAEs.
method Regularized Free-form Injective Flow (FIF) for flexible posterior.
result Full covariance VAEs outperform diagonal Gaussian posteriors.
Electronic power inverters are capable of quickly delivering reactive power to maintain customer voltages within operating tolerances and to reduce system losses in distribution grids. This paper proposes a systematic and data-driven approach to determine reactive power inverter output as a function of local measuremen…
Spatial processes with nonstationary and anisotropic covariance structure are often used when modelling, analysing and predicting complex environmental phenomena. Such processes may often be expressed as ones that have stationary and isotropic covariance structure on a warped spatial domain. However, the warping functi…
This work extends the randomized shortest paths (RSP) model by investigating the net flow RSP and adding capacity constraints on edge flows. The standard RSP is a model of movement, or spread, through a network interpolating between a random-walk and a shortest-path behavior [30, 42, 49]. The framework assumes a unit f…
Proposes a transformer model with geostatistical inductive bias for spatio-temporal forecasting.
problem Combining probabilistic rigor of geostatistics with flexible deep learning representations.
method Spatially-informed transformer with learnable covariance kernel.
result Successfully recovers spatial decay parameters end-to-end via backpropagation.
Holographic Invariant Storage uses vector architectures to ensure LLM safety at design time.
problem Mitigating context drift in large language models (LLMs) during deployment.
method Introduces Holographic Invariant Storage (HIS) protocol that combines known properties of bipolar Vector Symbolic Architectures into a design-time safety contract.
result Closed-form guarantees for single-signal recovery fidelity, continuous-noise robustness, and multi-signal capacity degradation are provided and validated.
SOC-ICNN expands neural network representational capacity by using conic optimization.
problem Restrictive representational capacity of ReLU-based ICNNs.
method Proposes SOC-ICNN architecture that uses Second-Order Cone Programming.
result SOC-ICNN strictly expands representational space without increasing complexity.
Researchers analyze backdoor data poisoning attacks and identify a memorization capacity parameter.
problem Understanding and mitigating backdoor data poisoning attacks in machine learning models.
method Formal theoretical framework, statistical and computational analysis, explicit constructions, and algorithm design.
result Identified a memorization capacity parameter to assess vulnerability to backdoor attacks and developed algorithms to detect and mitigate them.
Carbon capture and storage (CCS) can aid decarbonization of the atmosphere to limit further global temperature increases. A framework utilizing unsupervised learning is used to generate a range of subsurface geologic volumes to investigate potential sites for long-term storage of carbon dioxide. Generative adversarial …
How to model distribution of sequential data, including but not limited to speech and human motions, is an important ongoing research problem. It has been demonstrated that model capacity can be significantly enhanced by introducing stochastic latent variables in the hidden states of recurrent neural networks. Simultan…
Knowledge Distillation (KD) is a model-agnostic technique to improve model quality while having a fixed capacity budget. It is a commonly used technique for model compression, where a larger capacity teacher model with better quality is used to train a more compact student model with better inference efficiency. Throug…
A grand challenge in representation learning is to learn the different explanatory factors of variation behind the high dimen- sional data. Encoder models are often determined to optimize performance on training data when the real objective is to generalize well to unseen data. Although there is enough numerical eviden…
Injectivity of ReLU networks is characterized for generative models and inverse problems.
problem Injectivity in ReLU networks for generative models and inverse problems.
method Layerwise analysis, worst-case Lipschitz constants, differential topology, random projections.
result Global injectivity of ReLU networks requires expansivity between 3.4 and 10.5 for Gaussian matrices.
Classifies π1-injective maps between non-compact surfaces.
problem Characterizing maps with injective fundamental groups.
method Proper homotopy classification of maps.
result All π1-injective proper maps are classified. Positive injectivity radius for manifolds with Lie structure at infinity.
problem Injectivity radius positivity for manifolds with specific boundary conditions.
method Lie groupoids to prove injectivity radius positivity.
result Injectivity radius is positive for manifolds with Lie structure at infinity.
Geometric analysis improves noise injection in GANs.
problem Unclear mechanism of noise injection in GANs.
method Geometric framework based on Riemannian geometry.
result A new strategy for noise injection is devised.
The paper analyzes optimal dividend and capital injection strategies under time-inconsistent preferences.
problem Optimal dividend and capital injection strategies under time-inconsistent preferences.
method Diffusion risk model with general discount functions, weak equilibrium definition, HJB equation system.
result Explicit solutions and threshold types of optimal strategies derived under different discount functions.
A quasi-geodesic is Morse if and only if it is strongly contracting in injective spaces.
problem Characterizing Morse quasi-geodesics in injective spaces.
method Proving equivalence between Morse and strongly contracting quasi-geodesics.
result Injective metric spaces have the Morse local-to-global property and acylindrically hyperbolic groups with Morse elements.
This paper connects noise injection to Bayesian inference for neural networks, improving model uncertainty.
problem Improving the reliability and confidence of neural network predictions through uncertainty quantification.
method Introducing noise into neural network parameters during training and inference to estimate prediction uncertainty.
result The MCNI method outperforms baseline models in regression and classification tasks.
Compact theorem for minimal surfaces with lower injectivity radius.
problem Proving compactness of minimal surfaces with lower injectivity radius.
method Variant of Choi--Schoen compactness theorem, focusing on injectivity radius.
result Proved compactness theorem for minimal surfaces.
Lecture notes on group actions on injective spaces and Helly graphs.
problem Understanding group actions on specific metric spaces.
method Review of injective metric spaces and Helly graphs, elementary properties, constructions, and exercises.
result Presentation of various constructions of injective metric spaces and Helly graphs with interesting group actions.
Proves polynomial injectivity of Fubini-Study map for ample line bundles.
problem Injectivity of Fubini-Study map for ample line bundles.
method Polynomial injectivity proof with polynomial dependence on ample line bundle exponent.
result Quantitative version of injectivity proved, polynomial in ample line bundle exponent.
Lower bound on boundary injectivity radius for specific tubes.
problem Estimating the boundary injectivity radius of Margulis tubes.
method Using curvature bounds to derive a lower bound.
result A lower bound on the boundary injectivity radius is provided.
Reservoir subspace injection improves online ICA by preserving injected features.
problem Discarding injected features in top-n whitening can degrade performance. method Formalized reservoir subspace injection (RSI) and developed diagnostics (IER, SSO, ρ_x) to identify and mitigate the failure mode.
result RSI controller preserves passthrough retention, improving performance by up to 2.2 dB.
Injective and surjective neural operators for function spaces.
problem Tackles injective and surjective neural operators in function spaces.
method Combines prior work in ReLU and operator learning, uses Fredholm theory and Leray-Schauder degree theory.
result Injective and surjective neural operators are universal approximators and maintain their properties in finite-rank implementations.
Proposes a method to enhance graph models by injecting unseen connections.
problem Enhancing graph models to utilize unseen connections.
method Parametric link injection layer to find and inject weak connections.
result Improves performance on node classification and link prediction tasks.
The paper proves that certain spaces are injective and Helly graphs.
problem Understanding the structure of certain geometric and algebraic spaces.
method Building Helly graphs and injective metric spaces from lattices.
result The natural piecewise ℓ∞ metric on Euclidean buildings and Deligne complexes is injective. Smooths metrics on manifolds with curvature bounds and injectivity radius constraints.
problem Smooth metrics on manifolds with curvature and injectivity constraints.
method Bi-Lipschitz smoothing with controlled smoothing and volume lower bounds.
result Proves existence of smooth metrics with curvature bounds and injectivity radius constraints.
Nonuniform tubular neighborhoods of curves in Euclidean n-space are studied by using weighted distance functions and generalizing the normal exponential map. Different notions of injectivity radii are introduced to investigate singular but injective exponential maps. A generalization of the thickness formula is obtaine…
In this paper, we obtain two-sided bounds for the volumes of the Aloff-Wallach spaces W(p,q), compute maximal and minimal sectional curvature for the spaces W(n,n+1), and use this information to estimate the injectivity radii: We derive an upper bound for the injectivity radii of W(p,q) and a lower bound for the …
Injectivity of geodesic ray transform on specific Finsler manifolds proven.
problem Injectivity of geodesic ray transform on spherically symmetric reversible Finsler manifolds.
method Reduction to invertibility of generalized Abel transforms using angular Fourier series and Taylor expansions of geodesics.
result Injectivity of geodesic ray transform proven on specified Finsler manifolds.
Noise Injection probes deep learning dynamics during training phases.
problem Understanding the learning mechanism of deep neural networks.
method Noise Injection Nodes (NINs) are used to perturb DNN architectures without altering the optimization algorithm.
result Distinct training phases are observed based on the scale of injected noise.
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.
Generalizes a soul-bound for noncompact Alexandrov spaces.
problem Finding a lower bound for injectivity radius in Alexandrov spaces.
method Introduces the soul of Alexandrov spaces and applies a generalized bound.
result Injectivity radius is at least πK⁻¹/² if not equal to the soul's.
A labeled oriented tree is called injective if each generator occurs at most once as an edge label. We show that injective labeled oriented trees are aspherical. The proof relies on a new relative asphericity test based on a lemma of Stallings.
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.
Optimal control problem for firm cash flow with dividend and capital injection strategies.
problem Maximizing dividends while managing capital injections in a firm's cash flow.
method Proved two optimal strategies: mean-reverting dividends with capital injections or no injections until ruin.
result Optimal strategies are dichotomous: either mean-reverting dividends with injections or no injections.
The exponential map fails to be injective near critical points in sub-Riemannian geometry.
problem Injectivity failure of the exponential map at critical points in sub-Riemannian geometry.
method Analysis of the Hilbert invariant integral of the variational problem associated with the sub-Riemannian structure.
result Characterization of conjugate points in terms of metric structure.
Word metrics from large balls in injective spaces are exponentially generic and growth tight.
problem Understanding genericity and growth in word metrics from injective spaces.
method Geometric arguments and large ball considerations.
result Exponential genericity and growth tightness for word metrics.
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.
Introduces injective category number for continuous maps, linking classical and contemporary research.
problem Understanding conditions for a continuous map to be injective.
method Defines injective category number and examines its behavior under various operations.
result Provides a cohomological lower bound and expressions for injective category numbers in specific cases.
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…
It has been shown that injecting noise into the neural network weights during the training process leads to a better generalization of the resulting model. Noise injection in the distributed setup is a straightforward technique and it represents a promising approach to improve the locally trained models. We investigate…
We give lower bounds on the maximal injectivity radius for a closed orientable hyperbolic 3-manifold M with first Betti number 2, under some additional topological hypotheses. A corollary of the main result is that if M has first Betti number 2 and contains no fibroid surface then its maximal injectivity radius exceeds…
Upper bound on Stiefel manifold's injectivity radius found.
problem Finding the maximum distance within which the Stiefel manifold remains injective.
method Exhibited conjugate points and calculated the minimum of geodesic lengths.
result Upper bound on Stiefel manifold's injectivity radius is conjectured to be equal to the injectivity radius.