This paper studies curve multiplication in a specific algebraic structure.
problem Multiplication in the Kauffman bracket skein algebra of the thickened four-holed sphere.
method Developed an algorithm and explicit formula for curve multiplication, conjectured existence of a positive basis.
result Quasi-polynomial growth of the algorithm with respect to the number of crossings.
Sharp upper bound for quasi polynomial degree of manifold configuration spaces.
problem Determining the exact degree of quasi-polynomial homology groups of configuration spaces.
method Analyzing extremal homology groups of unordered configuration spaces of manifolds.
result The upper bound for the degree of quasi-polynomials is sharp for every manifold.
Algorithm distinguishes Gaussian mixtures from pure Gaussians in quasi-polynomial time.
problem Distinguishing mixtures of Gaussian components from pure Gaussians, especially when components are well-separated.
method Sum-of-Squares method, quasi-polynomial time algorithm, bipartitioning sample to separate components.
result Algorithm can reliably distinguish between mixtures and pure Gaussians in quasi-polynomial time.
Robust learning mixtures of linear regressions improve robustness.
problem Improving robustness in learning mixtures of linear regressions.
method Connecting mixtures of linear regressions and mixtures of Gaussians with thresholding for a quasi-polynomial time algorithm.
result The algorithm has significantly better robustness than previous results.
Efficient algorithm for identifying causal effects in linear models.
problem Determining causal effects from observational data under latent confounding.
method Symbolic computation and efficient algorithm for finding identifying formulas.
result Proves the existence of identifying formulas of a specified degree in quasi-polynomial time.
The paper establishes a nearly-sharp statistical threshold for efficient learning in Latent MDPs with separated components.
problem Learning Latent Markov Decision Processes (LMDPs) with separated components.
method The paper considers various notions of separation and establishes a nearly-sharp statistical threshold for efficient learning. It also presents a quasi-polynomial algorithm with time complexity scaling in terms of the statistical threshold under a weaker assumption of separability under the optimal policy, and a near-matching time complexity lower bound under the exponential time hypothesis.
result Establishes a nearly-sharp statistical threshold for efficient learning in Latent MDPs with separated components.
A sequence of rational functions in a variable q is q-holonomic if it satisfies a linear recursion with coefficients polynomials in q and qn. We prove that the degree of a q-holonomic sequence is eventually a quadratic quasi-polynomial. Our proof uses differential Galois theory (adapting proofs regarding hol…
In this paper we consider an elementary, and largely unexplored, combinatorial problem in low-dimensional topology. Consider a real 2-dimensional compact surface S, and fix a number of points F on its boundary. We ask: how many configurations of disjoint arcs are there on S whose boundary is F? We find that thi…
New algorithm for MDS with quasi-polynomial dependency on aspect ratio.
problem Finding an embedding that minimizes a specific objective function for given dissimilarities.
method A novel geometry-aware analysis of a conditional rounding of the Sherali-Adams LP hierarchy.
result Achieved a solution with cost \(O(\log Δ) \cdot extrm{OPT}^{Ω(1)} + ε\) in quasi-polynomial time.
Let S be an orientable surface with negative Euler characteristic. For k∈N, let Ck(S) denote the k-curve graph, whose vertices are isotopy classes of essential simple closed curves on S, and whose edges correspond to pairs of curves that can be realized to intersect at most …
New algorithm for learning mixtures with mostly uniform weights, improving on previous bounds.
problem Learning mixtures of Gaussians with uniform weights and mostly uniform component weights.
method Statistical Query (SQ) lower bound and quasi-polynomial upper bound for testing.
result Quasi-polynomial upper bound for testing mixtures with mostly uniform weights.
Counting essential surfaces in 3-manifolds yields concise formulae and detailed asymptotics.
problem Counting isotopy classes of essential surfaces in 3-manifolds.
method Normal and almost normal surfaces, Ehrhart's lattice point counting, ideal triangulations, and new essential surface testing.
result Quasi-polynomial behavior of surface counts and concise formulae for surface numbers.
It is well known that Sparse PCA (Sparse Principal Component Analysis) is NP-hard to solve exactly on worst-case instances. What is the complexity of solving Sparse PCA approximately? Our contributions include: 1) a simple and efficient algorithm that achieves an n−1/3-approximation; 2) NP-hardness of approximatio…
Proves quasimodularity of generating functions for pillowcase covers.
problem Counting Feynman-like graphs associated with quadratic differentials.
method Analyzing decompositions of half-translation surfaces into horizontal cylinders.
result Alternative proof of quasimodularity results and practical method to compute area Siegel-Veech constants.
Algorithm learns halfspaces with Tsybakov noise in polynomial time.
problem PAC learning halfspaces with adversarial noise.
method Reduction to certifying non-optimality, iterative process, warm-start algorithm.
result First polynomial-time algorithm for learning halfspaces with Tsybakov noise.
In this article Ehrhart quasi-polynomials of simplices are employed to determine isospectral lens spaces in terms of a finite set of numbers. Using the natural lattice associated with a lens space the associated toric variety of a lens space is introduced. It is proved that if two lens spaces are isospectral then the d…
Sparse PCA algorithm improves upon existing methods with better guarantees.
problem Recovering sparse vectors from Gaussian samples with adversarial perturbations.
method New algorithm running in polynomial time with improved β threshold.
result Better guarantees than Covariance Thresholding for large t.
We give an algorithm for completing an order-m symmetric low-rank tensor from its multilinear entries in time roughly proportional to the number of tensor entries. We apply our tensor completion algorithm to the problem of learning mixtures of product distributions over the hypercube, obtaining new algorithmic result…
Cylindrical contact homology linked to Ehrhart polynomials and Chen-Ruan cohomology.
problem Contact invariants of Q-Gorenstein toric contact manifolds.
method Relationships between cylindrical contact homology and Ehrhart polynomials, Chen-Ruan cohomology.
result Cylindrical contact homology invariants linked to Ehrhart polynomials and Chen-Ruan cohomology.
Tensor rank and low-rank tensor decompositions have many applications in learning and complexity theory. Most known algorithms use unfoldings of tensors and can only handle rank up to n⌊p/2⌋ for a p-th order tensor in Rnp. Previously no efficient algorithm can decompose 3rd order ten…
Abstract Coxeter groups have growth rates that are Perron numbers.
problem Understanding growth rates of Coxeter groups.
method Defined a class of Coxeter groups, ∞--spanned, and analyzed their growth rates. result For ∞--spanned Coxeter groups, geodesic growth rate strictly dominates word growth rate and appears to be a Perron number. Historical economic growth in countries of the former USSR is analysed. It is shown that Unified Growth Theory is contradicted by the data, which were used, but not analysed, during the formulation of this theory. Unified Growth Theory does not explain the mechanism of economic growth. It explains the mechanism of Malt…
Study measures economic growth sources in Iran's mining sector using neoclassical growth accounting.
problem Determining the share of economic growth sources in Iran's mining sector.
method Neoclassical growth accounting approach, using production function and Solow residual equation.
result Average annual growth rate of TFP was 2.94% over 30 years.
Simple math predicts growth trends.
problem Complex growth projections are hard to understand.
method Direct or indirect analysis of growth rates.
result Simple assumptions lead to understandable growth predictions.
Unified Growth Theory debunked: economic growth is insecure and unsustainable.
problem The mystery of the great divergence in income per capita.
method Analysis of economic data to show that growth trajectories are increasing vertically over time.
result Unified Growth Theory is incorrect and promotes misleading concepts.
Calculates the systolic growth of nilpotent Lie groups, providing new insights.
problem Understanding the growth of systolic volume in nilpotent Lie groups.
method Expressed systolic growth in terms of discrete subrings, developed methods for lower bounds.
result First computations of systolic growth for non-equivalent volume growth cases.
The Unified Growth Theory is a puzzling collection of myths based on illusions created by hyperbolic distributions. Some of these myths are discussed. The examination of data shows that the three stages of growth (Malthusian Regime, Post-Malthusian Regime and Modern Growth Regime) did not exist and that Industrial Revo…
New method matches networks faster and more accurately than previous approaches.
problem Matching networks with smooth structures efficiently.
method Graphon estimation and sampling-and-matching scheme.
result Consistent polynomial-time solution for unseeded graph matching.
Historical economic growth in Asia (excluding Japan) is analysed. It is shown that Unified Growth Theory is contradicted by the data, which were used (but not analysed) during the formulation of this theory. Unified Growth Theory does not explain the mechanism of economic growth. It explains the mechanism of Malthusian…
Study confined subgroups in groups with contracting elements, showing their growth rate is strictly greater than half of the ambient growth rate.
problem Understanding the growth rate of confined subgroups in groups with contracting elements.
method Through boundary actions, analyzing the Hopf decomposition and quotient growth.
result Confined subgroups have a growth rate strictly greater than half of the ambient growth rate.
Growth rate of the world Growth Domestic Product (GDP) is analysed to determine possible pathways of the future economic growth. The analysis is based on using the latest data of the World Bank and it reveals that the growth rate between 1960 and 2014 was following a trajectory approaching asymptotically a constant val…
Novel parallelization simplifies machine learning algorithms.
problem Adapting machine learning algorithms to growing data and needs.
method A novel parallelization scheme that applies to broad learning algorithms.
result Reduces runtime to polylogarithmic time on quasi-polynomially many units.
Historical economic growth in Latin America is analysed using the data of Maddison. Unified Growth Theory is found to be contradicted by these data in the same way as it is contradicted by the economic growth in Africa, Asia, former USSR, Western Europe, Eastern Europe and by the world economic growth. Paradoxically, U…
Introduces G-Tutte polynomials for abelian group arrangements.
problem Counting homomorphisms from abelian groups to G. method Defines G-Tutte polynomials as a generalization of various polynomials. result Carries topological and enumerative information of abelian Lie group arrangements.
Growth rates of geodesics on modular orbifolds are studied.
problem Understanding growth rates of geodesics on modular orbifolds.
method Exhaustion of modular orbifold by compact subsurfaces, analysis of low lying geodesics and reciprocal geodesics.
result Growth rates of low lying geodesics and reciprocal geodesics converge to the full set's growth rate.
Data describing historical economic growth are analysed. Included in the analysis is the world and regional economic growth. The analysis demonstrates that historical economic growth had a natural tendency to follow hyperbolic distributions. Parameters describing hyperbolic distributions have been determined. A search …
Galor's mysterious income growth rate is debunked, revealing data manipulation.
problem Mysterious sudden spurt in income per capita growth rate.
method Mathematical analysis of historical world economic growth data.
result The sudden spurt in income per capita growth rate is an artifact of data presentation.
The study provides volume growth estimates for specific types of manifolds.
problem Estimating volume growth for Ricci solitons and quasi-Einstein manifolds.
method Similar to classical results, the study proves volume growth estimates for gradient Ricci solitons and quasi-Einstein manifolds.
result Sharp volume growth estimates for gradient shrinking Ricci solitons and upper bound volume growth estimates for quasi-Einstein manifolds.
Study growth rates of subgroups in groups with a constricting element.
problem Understanding growth rates of subgroups in groups with a constricting element.
method Examining the spectrum of relative and quotient exponential growth rates of quasi-convex subgroups.
result Determine when growth rates of subgroups are strictly smaller or coincide with the group's growth rate.
We present a quantitative characterisation of the fluctuations of the annualized growth rate of the real US GDP per capita growth at many scales, using a wavelet transform analysis of two data sets, quarterly data from 1947 to 2015 and annual data from 1800 to 2010. Our main finding is that the distribution of GDP grow…
Paper reviews and proves volume growth estimates for different types of gradient Ricci solitons.
problem Estimating volume growth for gradient Ricci solitons.
method Survey and prove new volume growth estimates.
result New volume growth estimates for expanding gradient Ricci solitons.
The paper studies stability and area growth of λ-hypersurfaces.
problem Stability and growth of area for λ-hypersurfaces. method Defined a F-functional and studied F-stability. result Lower and upper bounds for area growth of λ-hypersurfaces. New insights into groups with uniform exponential growth.
problem Uniform exponential growth in hierarchically hyperbolic groups.
method Quasi-isometric characterization and new insights into group structure.
result Uniform exponential growth for hierarchically hyperbolic groups.
Study groups with polynomial growth, finding structure and applications.
problem Understanding groups with polynomial growth structure.
method Structure theorem for locally compact groups of polynomial growth.
result Applications on various growth functions and relations to FC-G series.
In many European countries the growth of the real GDP per capita has been linear since 1950. An explanation for this linearity is still missing. We propose that in artificial intelligence we may find models for a linear growth of performance. We also discuss possible consequences of the fact that in systems with linear…
Study ancient caloric functions on graphs, extending a theorem from manifolds.
problem Bounding the dimension of ancient caloric functions on graphs.
method Extending Colding and Minicozzi's theorem to graphs.
result Dimension of ancient caloric functions is bounded by growth degree and graph dimension.
Optimal bounds found for ancient caloric functions on manifolds.
problem Bounding the dimension of ancient caloric functions on manifolds with polynomial volume growth.
method Analyzing polynomial growth and using Yau's conjecture for harmonic functions.
result Sharp bound for the dimension of ancient caloric functions on spaces where Yau's conjecture holds.
New proof shows not all Salem numbers are growth rates of Coxeter groups.
problem Identifying growth rates of Coxeter groups using Salem numbers.
method New proof using spectral radii and Coxeter transformations.
result Not every Salem number is a growth rate of hyperbolic Coxeter groups.