arXiv subjects

Sabyasachi Chatterjee

Publications and source records attributed to Sabyasachi Chatterjee.

At least 19 recordsLinked to original sources

arXiv2026

Averaging Molecular Dynamics simulations to study the slow-strain rate behavior of metals

The application of molecular dynamics (MD) simulations to quasistatic loading is severely limited by the large separation between atomic vibration timescales and experimentally relevant deformation rates. In this work we employ the Practical Time Averaging (PTA) framework to overcome this limitation and enable atomistic simulations of crystalline solids under quasistatic loading conditions. PTA exploits the intrinsic separation of timescales by defining slow variables as time-averaged observables of the fast atomistic dynamics and their evolution on the slow loading timescale, thereby avoiding explicit integration of the fast dynamics. Using this approach, we simulate uniaxial deformation, in both tension and compression, of 4 to 20 nm cubic specimens of face centered cubic aluminum nanocrystals at applied strain rates approaching quasistatic conditions. We define slow variables as the averaged kinetic energy, potential energy, and normal stress in the loading direction, and track their evolution on the slow timescale. The stress-strain curves show yielding close to the theoretical stress for homogeneous nucleation, followed by successive load drops and rises caused by dislocation nucleation, motion, and exit from free surfaces. The "smaller is harder" effect is evident from the stress-strain response and from the variation of yield stress with sample size. Serrations in the response are more pronounced for smaller samples. The effects of applied strain rate and initial temperature are also studied. The method also captures the evolution of intricate dislocation microstructures on the slow timescale by tracking time-averaged atomic positions. The PTA framework enables simulations at strain rates several orders of magnitude lower than those accessible to conventional MD, demonstrating significant speedup in computational time while retaining full atomistic resolution.

cond-mat.mtrl-sci↗

arXiv2026

Studying Creep-Fatigue interaction of Nickel-Based Superalloys using Crystal Plasticity and Entropy-Based life prediction model

Creep-fatigue interaction in single-crystal nickel superalloys is difficult to predict because the response depends on the combined effects of loading parameters, hold time, temperature, and the underlying deformation mechanisms. This is important for turbine blade applications, where components experience both fatigue and creep during service. In the present work, a crystal plasticity finite element (CPFE) framework is used to study the creep-fatigue response of a single-crystal nickel superalloy under a range of practically relevant thermo-mechanical loading conditions. In particular, the effects of strain amplitude, R-ratio, hold duration, and temperature on cyclic deformation, stress relaxation, damage evolution, and creep-fatigue life are examined. Particular attention is given to separate the roles of fatigue and creep damage, understanding their interaction, and identify the creep-dominated and fatigue-dominated regimes as a function of strain amplitude and hold time. The study brings together these effects within a single framework and shows that the predicted trends in cyclic response and life are in good agreement with experimental observations reported in the literature.

cond-mat.mtrl-sci↗

arXiv2026

An Exact Pointwise Characterization for Total Variation Denoising in Quantile Regression

Total variation denoising (TVD) is a classical method for denoising and curve fitting, yet an explicit pointwise description of its fitted values has only recently been established in the mean regression setting by arXiv:2410.03041v4. This raises the question of whether a similar representation holds for quantile regression. We answer this question affirmatively by deriving an exact minmax/maxmin representation for the quantile TVD estimator, providing a complete pointwise characterization of its solution set. Given that the quantile TVD estimator is generally non-unique, the existence of such a representation is perhaps surprising. We show that the set of admissible fitted values at any location forms a compact interval, whose endpoints are characterized exactly by minmax/maxmin functionals of local order statistics over nested intervals. We next develop several structural properties of the quantile TVD solution set. First, the solution set is closed under coordinatewise maximum and minimum, guaranteeing the existence of extremal elements -- upper and lower envelope solutions. Second, this reveals that quantile TVD is intrinsically non-crossing across quantile levels when a common tuning parameter is used. We prove this is driven by submodularity of the total variation penalty, and show that any penalized quantile regression estimator with a submodular penalty enjoys this property. From an estimation error perspective, our representation enables a refined pointwise analysis via a transparent local bias-variance decomposition, facilitating new pointwise risk bounds and near-optimal rates for locally Holder smooth functions. Our results hold under heavy-tailed noise (e.g., Cauchy) and substantially extend existing guarantees beyond locally constant signals. Altogether, these results advance the theory of quantile TV regression via exact pointwise min-max representations.

math.ST↗

arXiv2026

Minmax Trend Filtering: Generalizations of Total Variation Denoising via a Local Minmax/Maxmin Formula

Total Variation Denoising (TVD) is a fundamental denoising and smoothing method. In this article, we identify a new local minmax/maxmin formula producing two estimators which sandwich the univariate TVD estimator at every point. Operationally, this formula gives a local definition of TVD as a minmax/maxmin of a simple function of local averages. Moreover we find that this minmax/maxmin formula is generalizeable and can be used to define other TVD like estimators. In this article we propose and study higher order polynomial versions of TVD which are defined pointwise lying between minmax and maxmin optimizations of penalized local polynomial regressions over intervals of different scales. These appear to be new nonparametric regression methods, different from usual Trend Filtering and any other existing method in the nonparametric regression toolbox. We call these estimators Minmax Trend Filtering (MTF). We show how the proposed local definition of TVD/MTF estimator makes it tractable to bound pointwise estimation errors in terms of a local bias variance like trade-off. This type of local analysis of TVD/MTF is new and arguably simpler than existing analyses of TVD/Trend Filtering. In particular, apart from minimax rate optimality over bounded variation and piecewise polynomial classes, our pointwise estimation error bounds also enable us to derive local rates of convergence for (locally) Holder Smooth signals. These local rates offer a new pointwise explanation of local adaptivity of TVD/MTF instead of global (MSE) based justifications.

math.ST↗

arXiv2025

Risk Bounds For Distributional Regression

This work examines risk bounds for nonparametric distributional regression estimators. For convex-constrained distributional regression, general upper bounds are established for the continuous ranked probability score (CRPS) and the worst-case mean squared error (MSE) across the domain. These theoretical results are applied to isotonic and trend filtering distributional regression, yielding convergence rates consistent with those for mean estimation. Furthermore, a general upper bound is derived for distributional regression under non-convex constraints, with a specific application to neural network-based estimators. Comprehensive experiments on both simulated and real data validate the theoretical contributions, demonstrating their practical effectiveness.

stat.ML↗

arXiv2025

A new approach to locally adaptive polynomial regression

Adaptive bandwidth selection is a fundamental challenge in nonparametric regression. This paper introduces a new bandwidth selection procedure inspired by the optimality criteria for $\ell_0$-penalized regression. Although similar in spirit to Lepski's method and its variants in selecting the largest interval satisfying an admissibility criterion, our approach stems from a distinct philosophy, utilizing criteria based on $\ell_2$-norms of interval projections rather than explicit point and variance estimates. We obtain non-asymptotic risk bounds for the local polynomial regression methods based on our bandwidth selection procedure which adapt (near-)optimally to the local Hölder exponent of the underlying regression function simultaneously at all points in its domain. Furthermore, we show that there is a single ideal choice of a global tuning parameter in each case under which the above-mentioned local adaptivity holds. The optimal risks of our methods derive from the properties of solutions to a new ``bandwidth selection equation'' which is of independent interest. We believe that the principles underlying our approach provide a new perspective to the classical yet ever relevant problem of locally adaptive nonparametric regression.

stat.ML↗

arXiv2023

Central Limit Theorem for Gram-Schmidt Random Walk Design

We prove a central limit theorem for the Horvitz-Thompson estimator based on the Gram-Schmidt Walk (GSW) design, recently developed in Harshaw et al.(2022). In particular, we consider the version of the GSW design which uses randomized pivot order, thereby answering an open question raised in the same article. We deduce this under minimal and global assumptions involving only the problem parameters such as the (sum) potential outcome vector and the covariate matrix. As an interesting consequence of our analysis we also obtain the precise limiting variance of the estimator in terms of these parameters which is smaller than the previously known upper bound. The main ingredients are a simplified skeletal process approximating the GSW design and concentration phenomena for random matrices obtained from random sampling using the Stein's method for exchangeable pairs.

math.ST↗

arXiv2023

A Cross Validation Framework for Signal Denoising with Applications to Trend Filtering, Dyadic CART and Beyond

This paper formulates a general cross validation framework for signal denoising. The general framework is then applied to nonparametric regression methods such as Trend Filtering and Dyadic CART. The resulting cross validated versions are then shown to attain nearly the same rates of convergence as are known for the optimally tuned analogues. There did not exist any previous theoretical analyses of cross validated versions of Trend Filtering or Dyadic CART. To illustrate the generality of the framework we also propose and study cross validated versions of two fundamental estimators; lasso for high dimensional linear regression and singular value thresholding for matrix estimation. Our general framework is inspired by the ideas in Chatterjee and Jafarov (2015) and is potentially applicable to a wide range of estimation methods which use tuning parameters.

math.ST↗

arXiv2022

Spatially Adaptive Online Prediction of Piecewise Regular Functions

We consider the problem of estimating piecewise regular functions in an online setting, i.e., the data arrive sequentially and at any round our task is to predict the value of the true function at the next revealed point using the available data from past predictions. We propose a suitably modified version of a recently developed online learning algorithm called the sleeping experts aggregation algorithm. We show that this estimator satisfies oracle risk bounds simultaneously for all local regions of the domain. As concrete instantiations of the expert aggregation algorithm proposed here, we study an online mean aggregation and an online linear regression aggregation algorithm where experts correspond to the set of dyadic subrectangles of the domain. The resulting algorithms are near linear time computable in the sample size. We specifically focus on the performance of these online algorithms in the context of estimating piecewise polynomial and bounded variation function classes in the fixed design setup. The simultaneous oracle risk bounds we obtain for these estimators in this context provide new and improved (in certain aspects) guarantees even in the batch setting and are not available for the state of the art batch learning estimators.

math.ST↗

arXiv2022

Element-wise Estimation Error of Generalized Fused Lasso

The main result of this article is that we obtain an elementwise error bound for the Fused Lasso estimator for any general convex loss function $ρ$. We then focus on the special cases when either $ρ$ is the square loss function (for mean regression) or is the quantile loss function (for quantile regression) for which we derive new pointwise error bounds. Even though error bounds for the usual Fused Lasso estimator and its quantile version have been studied before; our bound appears to be new. This is because all previous works bound a global loss function like the sum of squared error, or a sum of Huber losses in the case of quantile regression in Padilla and Chatterjee (2021). Clearly, element wise bounds are stronger than global loss error bounds as it reveals how the loss behaves locally at each point. Our element wise error bound also has a clean and explicit dependence on the tuning parameter $λ$ which informs the user of a good choice of $λ$. In addition, our bound is nonasymptotic with explicit constants and is able to recover almost all the known results for Fused Lasso (both mean and quantile regression) with additional improvements in some cases.

math.ST↗

arXiv2022

Localising change points in piecewise polynomials of general degrees

In this paper we are concerned with a sequence of univariate random variables with piecewise polynomial means and independent sub-Gaussian noise. The underlying polynomials are allowed to be of arbitrary but fixed degrees. All the other model parameters are allowed to vary depending on the sample size. We propose a two-step estimation procedure based on the $\ell_0$-penalisation and provide upper bounds on the localisation error. We complement these results by deriving a global information-theoretic lower bounds, which show that our two-step estimators are nearly minimax rate-optimal. We also show that our estimator enjoys near optimally adaptive performance by attaining individual localisation errors depending on the level of smoothness at individual change points of the underlying signal. In addition, under a special smoothness constraint, we provide a minimax lower bound on the localisation errors. This lower bound is independent of the polynomial orders and is sharper than the global minimax lower bound.

math.ST↗

arXiv2021

Regret Minimization in Isotonic, Heavy-Tailed Contextual Bandits via Adaptive Confidence Bands

In this paper we initiate a study of non parametric contextual bandits under shape constraints on the mean reward function. Specifically, we study a setting where the context is one dimensional, and the mean reward function is isotonic with respect to this context. We propose a policy for this problem and show that it attains minimax rate optimal regret. Moreover, we show that the same policy enjoys automatic adaptation; that is, for subclasses of the parameter space where the true mean reward functions are also piecewise constant with $k$ pieces, this policy remains minimax rate optimal simultaneously for all $k \geq 1.$ Automatic adaptation phenomena are well-known for shape constrained problems in the offline setting; %The phenomenon of automatic adaptation of shape constrained methods is known to occur in offline problems; we show that such phenomena carry over to the online setting. The main technical ingredient underlying our policy is a procedure to derive confidence bands for an underlying isotonic function using the isotonic quantile estimator. The confidence band we propose is valid under heavy tailed noise, and its average width goes to $0$ at an adaptively optimal rate. We consider this to be an independent contribution to the isotonic regression literature.

math.ST↗

arXiv2021

Quantile Regression by Dyadic CART

In this paper we propose and study a version of the Dyadic Classification and Regression Trees (DCART) estimator from Donoho (1997) for (fixed design) quantile regression in general dimensions. We refer to this proposed estimator as the QDCART estimator. Just like the mean regression version, we show that a) a fast dynamic programming based algorithm with computational complexity $O(N \log N)$ exists for computing the QDCART estimator and b) an oracle risk bound (trading off squared error and a complexity parameter of the true signal) holds for the QDCART estimator. This oracle risk bound then allows us to demonstrate that the QDCART estimator enjoys adaptively rate optimal estimation guarantees for piecewise constant and bounded variation function classes. In contrast to existing results for the DCART estimator which requires subgaussianity of the error distribution, for our estimation guarantees to hold we do not need any restrictive tail decay assumptions on the error distribution. For instance, our results hold even when the error distribution has no first moment such as the Cauchy distribution. Apart from the Dyadic CART method, we also consider other variant methods such as the Optimal Regression Tree (ORT) estimator introduced in Chatterjee and Goswami (2019). In particular, we also extend the ORT estimator to the quantile setting and establish that it enjoys analogous guarantees. Thus, this paper extends the scope of these globally optimal regression tree based methodologies to be applicable for heavy tailed data. We then perform extensive numerical experiments on both simulated and real data which illustrate the usefulness of the proposed methods.

stat.ME↗

arXiv2021

Risk Bounds for Quantile Trend Filtering

We study quantile trend filtering, a recently proposed method for nonparametric quantile regression with the goal of generalizing existing risk bounds known for the usual trend filtering estimators which perform mean regression. We study both the penalized and the constrained version (of order $r \geq 1$) of univariate quantile trend filtering. Our results show that both the constrained and the penalized version (of order $r \geq 1$) attain the minimax rate up to log factors, when the $(r-1)$th discrete derivative of the true vector of quantiles belongs to the class of bounded variation signals. Moreover we also show that if the true vector of quantiles is a discrete spline with a few polynomial pieces then both versions attain a near parametric rate of convergence. Corresponding results for the usual trend filtering estimators are known to hold only when the errors are sub-Gaussian. In contrast, our risk bounds are shown to hold under minimal assumptions on the error variables. In particular, no moment assumptions are needed and our results hold under heavy-tailed errors. Our proof techniques are general and thus can potentially be used to study other nonparametric quantile regression methods. To illustrate this generality we also employ our proof techniques to obtain new results for multivariate quantile total variation denoising and high dimensional quantile linear regression.

math.ST↗

arXiv2021

A discrete dislocation dynamics study of precipitate bypass mechanisms in nickel-based superalloys

Order strengthening in nickel-based superalloys is associated with the extra stress required for dislocations to bypass the $γ'$ precipitates distributed in the $γ$ matrix. A rich variety of bypass mechanism has been identified, with various shearing and Orowan looping processes giving way to climb bypass as the operating conditions change from the low/intermediate temperatures and high stress regime, to the high temperature and low stress regime. When anti phase boundary (APB) shearing and Orowan looping mechanisms operate, the bypass mechanism changes from shearing to looping with increased particle size and within a broad coexistence size window. Another possibility, supported by indirect experimental evidence, is that a third "hybrid" transition mechanism may operate. In this paper we use discrete dislocation dynamics (DDD) simulations to study dislocation bypass mechanisms in Ni-based superalloys. We develop a new method to compute generalized stacking fault forces in DDD simulations. We use this method to study the mechanisms of bypass of a square lattice of spherical $γ'$ precipitates by $a/2\langle110\rangle\{111\}$ edge dislocations, as a function of the precipitates volume fraction and size. We show that the hybrid mechanism is possible and it operates as a transition mechanism between the shearing and looping regimes over a large range of precipitates volume fraction and radii. We also consider the effects of a $γ/γ'$ lattice misfit on the bypass mechanisms, which we approximate by an additional precipitate stress computed according to Eshelby's inclusion theory. We show that in the shearing and hybrid looping-shearing regimes, a lattice misfit generally results in an increased bypass stress. For sufficiently high lattice misfit, the bypass stress is controlled by the pinning of the trailing dislocation on the exit side of the precipitates.

cond-mat.mtrl-sci↗

arXiv2021

New Risk Bounds for 2D Total Variation Denoising

2D Total Variation Denoising (TVD) is a widely used technique for image denoising. It is also an important nonparametric regression method for estimating functions with heterogenous smoothness. Recent results have shown the TVD estimator to be nearly minimax rate optimal for the class of functions with bounded variation. In this paper, we complement these worst case guarantees by investigating the adaptivity of the TVD estimator to functions which are piecewise constant on axis aligned rectangles. We rigorously show that, when the truth is piecewise constant, the ideally tuned TVD estimator performs better than in the worst case. We also study the issue of choosing the tuning parameter. In particular, we propose a fully data driven version of the TVD estimator which enjoys similar worst case risk guarantees as the ideally tuned TVD estimator.

math.ST↗

arXiv2021

Adaptive Estimation of Multivariate Piecewise Polynomials and Bounded Variation Functions by Optimal Decision Trees

Proposed by Donoho (1997), Dyadic CART is a nonparametric regression method which computes a globally optimal dyadic decision tree and fits piecewise constant functions in two dimensions. In this article we define and study Dyadic CART and a closely related estimator, namely Optimal Regression Tree (ORT), in the context of estimating piecewise smooth functions in general dimensions in the fixed design setup. More precisely, these optimal decision tree estimators fit piecewise polynomials of any given degree. Like Dyadic CART in two dimensions, we reason that these estimators can also be computed in polynomial time in the sample size $N$ via dynamic programming. We prove oracle inequalities for the finite sample risk of Dyadic CART and ORT which imply tight risk bounds for several function classes of interest. Firstly, they imply that the finite sample risk of ORT of order $r \geq 0$ is always bounded by $C k \frac{\log N}{N}$ whenever the regression function is piecewise polynomial of degree $r$ on some reasonably regular axis aligned rectangular partition of the domain with at most $k$ rectangles. Beyond the univariate case, such guarantees are scarcely available in the literature for computationally efficient estimators. Secondly, our oracle inequalities uncover minimax rate optimality and adaptivity of the Dyadic CART estimator for function spaces with bounded variation. We consider two function spaces of recent interest where multivariate total variation denoising and univariate trend filtering are the state of the art methods. We show that Dyadic CART enjoys certain advantages over these estimators while still maintaining all their known guarantees.

math.ST↗

arXiv2020

How useful are formal hierarchies? A case study on averaging dislocation dynamics to define meso-macro plasticity

A formal hierarchy of exact evolution equations are derived for physically relevant space-time averages of state functions of microscopic dislocation dynamics. While such hierarchies are undoubtedly of some value, a primary goal here is to expose the intractable complexity of such systems of nonlinear partial differential equations that, furthermore, remain `non-closed' and therefore subject to phenomenological assumptions to be useful. It is instead suggested that such hierarchies be terminated at the earliest stage possible and effort be expended to derive closure relations for the `non-closed' terms that arise from the formal averaging by taking into account the full-stress-coupled microscopic dislocation dynamics, a matter on which these formal hierarchies, whether of kinetic theory type or as pursued here, are silent.

cond-mat.mtrl-sci↗