Proves volume conjecture for twist knots using complex analysis.
problem Volume conjecture for twist knots.
method Equivalence relation, complex analysis, analytic continuation, function of several complex variables.
result Proves volume conjecture for twist knots.
Unified framework for complex, split-complex, and dual numbers.
problem Analytic and geometric scope of real-analytic functions.
method Generalized Cauchy-Riemann structure and unified real algebra family.
result Milnor-Le type fibration theorem for nondegenerate algebras.
The paper finds transformation formulas for quaternionic complex structures.
problem Quaternionic projective invariance of k-Cauchy-Fueter complex. method Explicit transformation formulae under mSL(n+1,H). result Quaternionic projectively invariant operator and defining density.
We study the complexity of the entire regularization path for least squares regression with 1-norm penalty, known as the Lasso. Every regression parameter in the Lasso changes linearly as a function of the regularization value. The number of changes is regarded as the Lasso's complexity. Experimental results using exac…
Improved sample complexity for Gaussian process approximations.
problem Efficiently approximating Gaussian processes with sparse spectrum.
method Improved sample complexity analysis and auto-encoding algorithm.
result Gaussian process predictions and model evidence can be well-approximated with low sample complexity.
The h-principle helps solve complex geometric problems.
problem Solving complex geometric problems using the h-principle.
method Developed from the Oka-Grauert principle and Gromov's theory, the h-principle is applied to Oka manifolds and maps.
result Recent developments and applications of the h-principle in complex analysis and geometry.
Paper tackles complex risk in deep neural networks.
problem Complex risk in deep neural networks.
method Developed new approach for complex risk statistics.
result Derived dual representation for complex risk.
We apply information-based complexity analysis to support vector machine (SVM) algorithms, with the goal of a comprehensive continuous algorithmic analysis of such algorithms. This involves complexity measures in which some higher order operations (e.g., certain optimizations) are considered primitive for the purposes …
The authors study the method of scaling in the context of the study of automorphism groups of complex domains in multiple dimensions. Various types of scaling techniques are compared and contrasted. Applications are given in a number of areas of complex geometric analysis. Relations with other parts of mathematics are …
Study local and global aspects of complex plane curve embeddings.
problem Local and global problems of complex plane curve embeddings.
method Braid monodromy, local and global analysis.
result Historical progress in understanding complex plane curve embeddings.
Smoothed analysis of complexity bounds and condition numbers has been done, so far, on a case by case basis. In this paper we consider a reasonably large class of condition numbers for problems over the complex numbers and we obtain smoothed analysis estimates for elements in this class depending only on geometric inva…
Paper introduces probabilistic methods to approximate archetypal analysis, reducing complexity.
problem Inherent computational complexity of archetypal analysis limits its practical applicability.
method Two preprocessing techniques: dimensionality reduction and representation cardinality reduction, using probabilistic geometry.
result The method effectively reduces scaling and provides near-optimal solutions for prediction errors.
New TDA approach using Finsler metrics.
problem Traditional TDA concepts and methods.
method Introducing Finsler metrics for TDA.
result Relevance of Finsler metrics to TDA.
Complex network analysis reveals dominant stocks in financial stock returns correlations.
problem Inferring financial stock returns correlations from complex network analysis.
method Simulated geometric Brownian motion for stocks, complex network analysis, eigenvector centrality, clustering.
result Returns correlation matrix is dominated by stocks with high eigenvector centrality and clustering.
Paper analyzes sample complexity of polynomial neural networks.
problem Understanding the sample complexity of polynomial neural networks.
method Extends previous literature to polynomial neural networks and analyzes sample complexity.
result Obtains novel results on sample complexity of polynomial neural networks.
The construction of synthetic complex-valued signals from real-valued observations is an important step in many time series analysis techniques. The most widely used approach is based on the Hilbert transform, which maps the real-valued signal into its quadrature component. In this paper, we define a probabilistic gene…
Deep model learns complex latent codes without assuming factor structure.
problem Learning latent codes with complex, non-factorial distributions.
method Deep generative factor analysis with beta process prior and stochastic EM algorithm.
result Preliminary results show model can approximate complex distributions.
The extragradient method accelerates convergence in complex game dynamics.
problem Complex interactions in game dynamics cause simple methods to diverge, necessitating more sophisticated approaches.
method A polynomial-based analysis to identify three scenarios for accelerated convergence of the momentum extragradient method.
result The momentum extragradient method achieves faster convergence under specific eigenvalue conditions.
We propose the labeled Čech complex, the plain labeled Vietoris-Rips complex, and the locally scaled labeled Vietoris-Rips complex to perform persistent homology inference of decision boundaries in classification tasks. We provide theoretical conditions and analysis for recovering the homology of a decision boundary fr…
Complex analysis aids in studying minimal surfaces.
problem Understanding minimal surfaces in Euclidean spaces.
method Complex-analytic techniques applied to conformal minimal surfaces.
result New results on approximation, interpolation, and general position properties.
Transformers show strengths and weaknesses in complexity analysis.
problem Understanding the strengths and limitations of attention layers in transformers.
method Analysis of representation power through complexity parameters and task-specific constructions.
result Transformers can solve sparse averaging tasks with logarithmic complexity, but triple detection tasks require linear complexity.
New complexity analysis for estimating normalizing constants in high dimensions.
problem Estimating the normalizing constant of unnormalized probability densities in high dimensions.
method Analyze and derive the oracle complexity of annealed importance sampling.
result Oracle complexity of $\widetilde{O}\left(\frac{dβ^2{\mathcal{A}}^2}{\varepsilon^4}
ight)$ for estimating Z within ε relative error. Complex analysis techniques link Gaussian RBF kernels to quantum mechanics.
problem Understanding the Gaussian RBF kernel in machine learning and SVMs.
method Using Fock space and Segal-Bargmann theories in complex analysis.
result Proves connections between Gaussian RBF kernels and quantum mechanics operators.
Deep learning improves survival analysis for complex data types.
problem Limited application of DL in survival analysis for complex data.
method Comprehensive review of DL methods for time-to-event analysis.
result Methods often ignore complex settings like multiple risks and censoring.
Study analyzes Echo State Network parameters for Rossler attractor dynamics.
problem Understanding the influence of network type on Echo State Network performance.
method Experimental analysis of Echo State Network parameters using Rossler attractor.
result Exploration of how network type affects Echo State Network performance.
Study compares Bitcoin, gold, and gas price complexity using multifractal and multiscale entropy methods.
problem Quantifying complexity of financial time series for market analysis.
method Employed MF-DFA and RCMSE to analyze Bitcoin, GBP/USD, gold, and natural gas price log-return time series.
result Bitcoin shows higher complexity compared to other markets, linked to higher nonlinear correlations.
Lyapunov-based analysis shows polynomial sample complexity for WCMDPs and RBs.
problem Learning in WCMDPs and RBs under a generative model.
method Lyapunov-based analysis framework.
result Near-optimal policies can be learned with polynomial complexity.
Paper introduces Simplet Frequency Distribution (SFD) for SCs.
problem Frequency analysis of simplets in large SCs.
method Developed SFD vector and uniform sampling-based algorithm.
result Validated theoretical bounds with experiments.
We introduce a simple analysis of the structural complexity of infinite-memory processes built from random samples of stationary, ergodic finite-memory component processes. Such processes are familiar from the well known multi-arm Bandit problem. We contrast our analysis with computation-theoretic and statistical infer…
We carry out a Painlevé analysis to find the cases where the cohomogeneity one steady Ricci soliton equation can be integrable. We concentrate on two classes of solitons: warped products and complex line bundles over a Fano Kähler Einstein base. For warped products, the analysis singles out the case with one factor whe…
The paper analyzes adversarial robustness for linear models and neural networks using Rademacher complexity.
problem Understanding adversarial robustness of linear models and neural networks.
method The paper uses Rademacher complexity to provide upper and lower bounds for adversarial robustness of linear hypotheses and neural networks.
result The paper provides bounds on adversarial Rademacher complexity for linear hypotheses and neural networks, offering a finer analysis of input dimensionality.
Analyzes the complexity of linear hypothesis sets using Rademacher complexity.
problem Understanding the complexity of linear hypothesis sets for various norms.
method Tight analysis of empirical Rademacher complexity for linear hypothesis classes with bounded weights.
result Improved bounds on Rademacher complexity for linear hypothesis sets, matching or improving existing results.
Study controls bifurcations in Eulerian flows with multiple Hopf singularities.
problem Bifurcation analysis and control of nonlinear Eulerian flows with non-resonant n-tuple Hopf singularities.
method Analysis of CW complex bifurcations of flow-invariant Clifford hypertori, using leaf-bifurcation varieties.
result Tertiary toral CW complex bifurcates from and persists outside a secondary toral CW complex.
Predicting the runtime complexity of a programming code is an arduous task. In fact, even for humans, it requires a subtle analysis and comprehensive knowledge of algorithms to predict time complexity with high fidelity, given any code. As per Turing's Halting problem proof, estimating code complexity is mathematically…
Classic economic science is reaching the limits of its explanatory powers. Complexity science uses an increasingly larger set of different methods to analyze physical, biological, cultural, social, and economic factors, providing a broader understanding of the socio-economic dynamics involved in the development of nati…
This study analyzes cryptocurrency market crashes using complex network analysis.
problem Identifying and understanding dynamics of cryptocurrency market crashes.
method Complex network analysis of cryptocurrency market during pre-crash, crash, and post-crash periods.
result Network density and clustering coefficient spike during crashes, indicating uninformed panic sell-off.
Geodesic convexity generalizes the notion of (vector space) convexity to nonlinear metric spaces. But unlike convex optimization, geodesically convex (g-convex) optimization is much less developed. In this paper we contribute to the understanding of g-convex optimization by developing iteration complexity analysis for …
SPCA improves PCA by learning from simple to complex samples.
problem Noise and outliers in complex data.
method Self-paced Principal Component Analysis (SPCA) that integrates samples from simple to more complex.
result SPCA improves state-of-the-art results on popular datasets.
No nontrivial automorphisms for cubic surfaces moduli space.
problem Understanding automorphisms of cubic surfaces moduli space.
method Analyzing the fundamental group of the moduli space.
result No nontrivial biholomorphic automorphisms for cubic surfaces moduli space.
A non-Hermitean extension of paradigmatic Wishart random matrices is introduced to set up a theoretical framework for statistical analysis of (real, complex and real quaternion) stochastic time series representing two "remote" complex systems. The first paper in a series provides a detailed spectral theory of non-Hermi…
In this paper, we present a unified analysis of matrix completion under general low-dimensional structural constraints induced by {\em any} norm regularization. We consider two estimators for the general problem of structured matrix completion, and provide unified upper bounds on the sample complexity and the estimatio…
Quantum Monte Carlo speeds up option pricing for complex payoff functions.
problem Efficiently pricing options with complex payoff functions using quantum computing.
method Developed a quantum Monte Carlo algorithm for multidimensional Black-Scholes PDEs.
result Proved polynomial computational complexity and speed-up over classical methods.
This paper analyzes sampling from heavy-tailed distributions using discretized Itô diffusions.
problem Sampling from heavy-tailed distributions with finite variance.
method Mean-square analysis of discretized Itô diffusions with weighted Poincaré inequalities.
result Explicit iteration complexity for obtaining samples close to target distributions in Wasserstein-2 metric.
Multifractality is ubiquitously observed in complex natural and socioeconomic systems. Multifractal analysis provides powerful tools to understand the complex nonlinear nature of time series in diverse fields. Inspired by its striking analogy with hydrodynamic turbulence, from which the idea of multifractality originat…
In this paper we give definitions of matrix rates of return which do not depend on the choice of basis describing baskets. We give their economic interpretation. The matrix rate of return describes baskets of arbitrary type and extends portfolio analysis to the complex variable domain. This allows us for simultaneous a…
This note provides a new proof of the real analyticity of the Liouville map.
problem Real analyticity of the Liouville map on Riemann surfaces.
method Complex analysis approach.
result Real analyticity of the Liouville map proved using complex analysis.
Performance monitoring, anomaly detection, and root-cause analysis in complex cyber-physical systems (CPSs) are often highly intractable due to widely diverse operational modes, disparate data types, and complex fault propagation mechanisms. This paper presents a new data-driven framework for root-cause analysis, based…
New analysis improves sample complexity for vanilla policy gradient methods.
problem Improving sample complexity guarantees for vanilla policy gradient methods.
method Adapting tools from SGD analysis to policy gradient methods, with smoothness and gradient approximation assumptions.
result Established improved sample complexity bounds for convergence and global optimum.