Piecewise polynomial interpolation-based gradient descent reduces oracle complexity for smooth loss functions.
problem Optimizing empirical risk minimization loss functions
method Piecewise polynomial interpolation-based gradient descent
result Oracle complexity is reduced for smooth loss functions
New method efficiently interpolates nonparametric density estimators.
problem Efficient evaluation of nonparametric density estimators.
method Piecewise multivariate polynomial interpolation scheme.
result New estimator with low space requirements and efficient querying.
Unified framework explains why overfitting is benign in interpolating learning.
problem Understanding why overfitting is benign in highly overparameterized models.
method Spectral-transport stability framework.
result Sharp benign-overfitting criterion and explicit phase-transition rates.
Neural networks can interpolate random data but still generalize well, studied in the NT regime.
problem Understanding how neural networks interpolate random labels and generalize well in the overparametrized regime.
method Characterization of the eigenstructure of the empirical NT kernel and generalization error of NT ridge regression.
result The generalization error is well approximated by polynomial ridge regression with an increased regularization parameter.
This is mainly a survey, explaining how the probabilistic (statistical mechanical) construction of Kahler-Einstein metrics on compact complex manifolds, introduced in a series of works by the author, naturally arises from classical approximation and interpolation problems in complex n-space. A fair amount of background…
Accelerates ERM problems with LPI-GD and improved oracle complexity.
problem Empirical Risk Minimization (ERM) problems with strong convexity and smoothness.
method Local Polynomial Interpolation-based Gradient Descent (LPI-GD) and accelerated methods.
result Oracle complexity improved to $ ilde{O}\left(\sqrtσ m^d \log(1/\varepsilon)
ight)$.
We conjecture a closed-form expression of HOMFLY-PT invariants of double twist knots colored by rectangular Young diagrams where the twist is encoded in interpolation Macdonald polynomials. We also put forth a conjecture of cyclotomic expansions of HOMFLY-PT polynomials colored by rectangular Young diagrams for any kno…
Proves a new law of robustness for interpolating arbitrary data distributions.
problem Understanding robust interpolation for arbitrary data distributions.
method Proves a Lipschitzness lower bound for robust interpolation.
result Demonstrates a two-fold law of robustness for interpolating functions.
Exact universal interpolation property for landmark configurations in Euclidean space.
problem Representing and deforming landmark configurations through flows of vector fields.
method Explicitly describe vector fields for exact universal interpolation property in all dimensions.
result Achieve controllability by combining constant and polynomial vector fields.
Deep networks can interpolate noisy data without losing generalization.
problem Characterizing the relationship between interpolation and generalization in overparameterized deep networks.
method Analyzing the loss landscape of neural network functions over volumes around training data points, varying model parameters and training epochs.
result Loss sharpness in the input space follows a double descent, with large models predicting noisy targets over larger volumes around training data points.
Private optimization faster on interpolation problems with quadratic growth.
problem Private optimization in interpolation problems.
method Adaptive algorithm with improved sample complexity.
result Exponential improvement in private sample complexity for quadratic growth.
We seek to improve the data efficiency of neural networks and present novel implementations of parameterized piece-wise polynomial activation functions. The parameters are the y-coordinates of n+1 Chebyshev nodes per hidden unit and Lagrangian interpolation between the nodes produces the polynomial on [-1, 1]. We show …
New basis for quantum gl_N invariants derived from Macdonald polynomials.
problem Constructing new bases for quantum gl_N invariants.
method Using interpolation Macdonald polynomials and Okounkov's results.
result Cyclotomic expansions for gl_N invariants and knot invariants.
A new tradeoff between regularization and sharpness improves model performance in overparameterized settings.
problem Improving model performance in overparameterized settings with minimum-norm interpolators.
method Proposes a regularization-sharpness tradeoff for overparameterized linear regression with an ℓ^p penalty.
result Empirical validation shows the tradeoff terms can distinguish performant linear interpolators.
Noise affects the effectiveness of interpolating models, especially those with strong inductive biases.
problem The impact of noise on interpolating models with strong inductive biases.
method Analyzing linear and classification models with sparse ground truths, proving fast rates for interpolators.
result Strong inductive biases can lead to faster but noisier interpolators, contrary to intuition.
In this paper we present an algorithm to reduce the area of a surface spanned by a finite number of boundary curves by initiating a variational improvement in the surface. The ansatz we suggest consists of original surface plus a variational parameter t multiplying the numerator H0 of mean curvature function def…
New algorithm interpolates data with neural nets, independent of sample size.
problem Understanding neural networks' ability to memorize training data.
method Randomized algorithm for constructing interpolating neural networks.
result Guarantees that are independent of the number of samples, moving beyond worst-case memorization capacity bounds.
Finite element method approximates scalar curvature in arbitrary dimensions.
problem Approximating scalar curvature using finite elements in arbitrary dimensions.
method Piecewise polynomial interpolants of a smooth Riemannian metric on a triangulated polyhedral domain.
result Finite element interpolants converge to scalar curvature with rate O(hr+1) in H−2(Ω) norm. Paper tackles blind polynomial regression for unknown inputs.
problem Fitting a polynomial to unknown or partially known input data.
method Formally defines the problem, proposes algorithmic approaches, and applies to jitter-correction.
result Proposes effective methods for blind polynomial regression.
Typically flat filling, linear or polynomial interpolation methods to generate missing historical data. We introduce a novel optimal method for recreating data generated by a diffusion process. The results are then applied to recreate historical data for stocks.
New law explains why deep learning models often have more parameters than needed.
problem Why deep learning models often have more parameters than classical theory suggests.
method Proved a universal law of robustness for smooth interpolation.
result Smooth interpolation requires d times more parameters than mere interpolation.
Globalizes Jones and Alexander polynomials using topological intersections.
problem Link invariants from graded intersections of Lagrangians.
method Topological model proving the Jones polynomial's well-definedness and constructing globalizations.
result Proves the Jones polynomial and constructs globalizations of Jones and Alexander polynomials.
The implied volatility is a crucial element of any financial toolbox, since it is used for quoting and the hedging of options as well as for model calibration. In contrast to the Black-Scholes formula its inverse, the implied volatility, is not explicitly available and numerical approximation is required. We propose a …
New 1-cocycles for knots identified via moduli spaces.
problem Identifying knots using topological moduli spaces and 1-cocycles.
method Upgrading Vassiliev invariant to combinatorial 1-cocycles and using Lagrange interpolation.
result Induces non-trivial pairing on knot homology groups.
Recurrent tasks such as pricing, calibration and risk assessment need to be executed accurately and in real-time. Simultaneously we observe an increase in model sophistication on the one hand and growing demands on the quality of risk management on the other. To address the resulting computational challenges, it is nat…
Sample- and computationally-efficient distribution estimation is a fundamental tenet in statistics and machine learning. We present SURF, an algorithm for approximating distributions by piecewise polynomials. SURF is: simple, replacing prior complex optimization techniques by straight-forward {empirical probability} ap…
Efficiently prices American options with multiple assets using sparse grids.
problem Pricing American options with multiple underlying assets efficiently.
method Dynamic programming formulation followed by sparse grid interpolation.
result Sparse grids reduce the number of interpolation points and maintain function smoothness.
In this paper, a rapid and high accurate numerical method for pricing discrete single and double barrier knock-out call options is presented. According to the well-known Black-Scholes framework, the price of option in each monitoring date could be calculate by computing a recursive integral formula upon the heat equati…
Lower bound proves ridgeless regression performs poorly near interpolation threshold.
problem Proving performance of ridgeless regression near interpolation threshold.
method Distribution-independent lower bound for mean squared error in noisy ridgeless linear regression.
result Lower bound implies ridgeless regression performs poorly near interpolation threshold.
Paper finds instantons for Kapustin-Witten equations on a specific manifold.
problem Existence of solutions to Kapustin-Witten equations on (0,∞)imesR2imesR. method Explains existence of solutions interpolating between two model solutions.
result Interpolation solutions exist with specific label constraints.
We prove twisted homological stability with polynomial coefficients for automorphism groups of free nilpotent groups of any given class. These groups interpolate between two extremes for which homological stability was known before, the general linear groups over the integers and the automorphism groups of free groups.…
Study on RF regression with SGD shows double descent phenomenon.
problem Understanding generalization in RF models trained with SGD.
method Precise non-asymptotic error bounds derived for RF regression under constant and polynomial-decay step-size SGD.
result RF regression generalizes well for interpolation learning and exhibits double descent behavior.
New quantum knot invariants derived from Verma modules.
problem Constructing universal quantum knot invariants from Verma modules.
method Defining level N universal invariants from finite quotients of Verma modules over quotient rings.
result Maximal universal invariants for prime N, interpolating Jones and ADO polynomials.
The study approximates option prices using Hermite polynomials without assuming a specific distribution.
problem Approximating option prices without assuming a specific distribution of returns.
method Approximating the logarithmic return's density by a linear combination of rescaled Hermite polynomials.
result Empirical results suggest reasonable performance for options with moderate strike prices.
New method uses higher-order Langevin dynamics for efficient parallel sampling.
problem Efficient parallel sampling from high-dimensional log-concave distributions.
method Combines higher-order Langevin dynamics with blockwise Lagrange polynomial interpolation.
result Reduces the number of parallel points required for a target accuracy.
Unified invariant of knots derived from Verma modules.
problem Constructing a unified invariant of knots from quantum sl2.
method Braid groups' action on tensors of Verma modules.
result Unified invariant interpolates colored Jones and ADO polynomials.
We prove constrained trace, matrix and constrained matrix Harnack inequalities for the nonlinear heat equation ωt=Δω+aωlnω on closed manifolds. We also derive a new interpolated Harnack inequality for the equation ωt=Δω−ωlnω+εRω on closed surfaces under the ε-Ricci flow. Finally we prove…
Let R be a real closed field, Q⊂R[Y1,...,Yℓ,X1,...,Xk], with $ °_{Y}(Q) \leq 2, °_{X}(Q) \leq d, Q \in {\mathcal Q}, #({\mathcal Q})=m$, and P⊂R[X1,...,Xk] with $°_{X}(P) \leq d, P \in {\mathcal P}, #({\mathcal P})=s$. Let S⊂Rℓ+k be a semi-alg…
The paper compares machine learning methods with traditional techniques for pricing and sensitivities of financial products with path-dependent structures.
problem Evaluating financial products with early-termination clauses, especially those with path-dependent structures.
method The paper compares regression methods including randomized recurrent and feed-forward neural networks, and a novel approach using signatures of the underlying price process, with traditional polynomial basis functions for pricing and sensitivities.
result Machine learning algorithms often match the accuracy and efficiency of traditional methods for Asian and look-back options, while randomized neural networks are best for callable certificates.
This paper shows how forward rate interpolations are equivalent to discount factor interpolations in yield curve construction.
problem The challenge of choosing between different interpolation methods for yield curve construction.
method Demonstrates the equivalence between forward rate interpolations and discount factor interpolations.
result Some popular interpolation methods on forward rates are equivalent to classical interpolation methods on discount factors.
By adopting the polynomial interpolation method, we propose an approach to hedge against the interest-rate risk of the default-free bonds by measuring the nonparallel movement of the yield-curve, such as the translation, the rotation and the twist. The empirical analysis shows that our hedging strategies are comparable…
Efficiently computes quasiconcave envelope with limited data.
problem Approximating unknown quasiconcave function with partial information.
method Solves value problem and interpolation problem with polynomial and logarithmic LPs.
result Efficiently computes quasiconcave envelope with limited data.
We consider the pseudo-Anosov elements of the mapping class group of a surface of genus g that fix a rank k subgroup of the first homology of the surface. We show that the smallest entropy among these is comparable to (k+1)/g. This interpolates between results of Penner and of Farb and the second and third authors, who…
New FX option interpolations impact implied volatilities.
problem Different interpolations of FX option quotes lead to varying implied volatilities.
method Analysis of various exact interpolations of broker quotes.
result Different interpolations result in different implied volatilities.
Kernel interpolation is inconsistent for norms with smoothness above a constant.
problem Inconsistency of kernel interpolation in reproducing kernel Hilbert spaces.
method Lower bounds for generalization error in Sobolev norms.
result Kernel interpolation is always inconsistent for norms with smoothness above a constant.
Bayesian interpolants explain neural network inferences concisely.
problem Understanding neural network inferences.
method Adapting Craig interpolants for neural networks.
result Produces precise, understandable explanations.
Near-interpolating models grow norms quickly, affecting generalization.
problem Understanding the trade-off between interpolation and generalization in near-interpolating models.
method Random matrix theory and eigendecay analysis of data covariance matrix.
result Near-interpolating models exhibit rapid norm growth and worse generalization trade-offs.
The paper improves interpolation in generative models by using specific base distributions.
problem Unexpected side effects in linear interpolations of normalizing flows.
method Enforces a specific manifold using Dirichlet and von Mises-Fisher base distributions.
result Superior performance in terms of bits per dimension, FID, and KID scores for interpolation.