arXiv subjects

Sabyasachi Chatterjee

Publications and source records attributed to Sabyasachi Chatterjee.

32 records · Page 2Linked to original sources

arXiv2020

Evaluation of Ultra Low Dose chest CT imaging for Covid 19 diagnosis and follow up

Objective: Computed Tomography (CT) has an important role to detect lung lesion related to Covide 19. The purpose of this work is to obtain diagnostic findings of Ultra-Low Dose (ULD) chest CT image and compare with routine dose chest CT. Material and Methods: Patients, suspected of Covid 19 infection, were scanned successively with routine dose, and ULD, with 98% or 94% dose reduction, protocols. Axial images of routine and ULD chest CT were evaluated objectively by two expert radiologists and quantitatively by Signal to Noise Ratio (SNR) and pixel by pixel noise measurement. Results: It was observed that the ULD and routine dose chest CT images could detect Covid 19 related lung lesions in patients with PCR positive test. Also, SNR and pixel noise values were comparable in these protocols. Conclusion: ULD chest CT with 98% dose reduction can be used in non-pandemic situation as a substitute for chest radiograph for screening and follow up. Routine chest CT protocol can be replaced by ULD, with 94% dose reduction, to detect patients suspected with Covid 19 at an early stage and for its follow up.

physics.med-ph↗

arXiv2020

Plasticity without phenomenology: a first step

A novel, concurrent multiscale approach to meso/macroscale plasticity is demonstrated. It utilizes a carefully designed coupling of a partial differential equation (pde) based theory of dislocation mediated crystal plasticity with time-averaged inputs from microscopic Dislocation Dynamics (DD), adapting a state-of-the-art mathematical coarse-graining scheme. The stress-strain response of mesoscopic samples at realistic, slow, loading rates up to appreciable values of strain is obtained, with significant speed-up in compute time compared to conventional DD. Effects of crystal orientation, loading rate, and the ratio of the initial mobile to sessile dislocation density on the macroscopic response, for both load and displacement controlled simulations are demonstrated. These results are obtained without using any phenomenological constitutive assumption, except for thermal activation which is not a part of microscopic DD. The results also demonstrate the effect of the internal stresses on the collective behavior of dislocations, manifesting, in a set of examples, as a Stage I to Stage II hardening transition.

cs.CE↗

arXiv2018

Adaptive Risk Bounds in Univariate Total Variation Denoising and Trend Filtering

We study trend filtering, a relatively recent method for univariate nonparametric regression. For a given positive integer $r$, the $r$-th order trend filtering estimator is defined as the minimizer of the sum of squared errors when we constrain (or penalize) the sum of the absolute $r$-th order discrete derivatives of the fitted function at the design points. For $r=1$, the estimator reduces to total variation regularization which has received much attention in the statistics and image processing literature. In this paper, we study the performance of the trend filtering estimator for every positive integer $r$, both in the constrained and penalized forms. Our main results show that in the strong sparsity setting when the underlying function is a (discrete) spline with few "knots", the risk (under the global squared error loss) of the trend filtering estimator (with an appropriate choice of the tuning parameter) achieves the parametric $n^{-1}$ rate, up to a logarithmic (multiplicative) factor. Our results therefore provide support for the use of trend filtering, for every $r$, in the strong sparsity setting.

math.ST↗

arXiv2018

Prediction Rule Reshaping

Two methods are proposed for high-dimensional shape-constrained regression and classification. These methods reshape pre-trained prediction rules to satisfy shape constraints like monotonicity and convexity. The first method can be applied to any pre-trained prediction rule, while the second method deals specifically with random forests. In both cases, efficient algorithms are developed for computing the estimators, and experiments are performed to demonstrate their performance on four datasets. We find that reshaping methods enforce shape constraints without compromising predictive accuracy.

stat.ML↗

arXiv2017

Computing Singularly Perturbed Differential Equations

A computational tool for coarse-graining nonlinear systems of ordinary differential equations in time is discussed. Three illustrative model examples are worked out that demonstrate the range of capability of the method. This includes the averaging of Hamiltonian as well as dissipative microscopic dynamics whose `slow' variables, defined in a precise sense, can often display mixed slow-fast response as in relaxation oscillations, and dependence on initial conditions of the fast variables. Also covered is the case where the quasi-static assumption in solid mechanics is violated. The computational tool is demonstrated to capture all of these behaviors in an accurate and robust manner, with significant savings in time. A practically useful strategy for initializing short bursts of microscopic runs for the accurate computation of the evolution of slow variables is also developed.

math.NA↗

arXiv2017

On Estimation in Tournaments and Graphs under Monotonicity Constraints

We consider the problem of estimating the probability matrix governing a tournament or linkage in graphs from incomplete observations, under the assumption that the probability matrix satisfies natural monotonicity constraints after being permuted in both rows and columns by some latent permutation. This condition is classical in the social sciences literature (see Fishburn(1973)) and has been studied in the statistics literature in recent work (see Chatterjee (2015) and Shah et al. (2016)). In this paper, we investigate in detail a natural estimator which bypasses the need to search over all possible latent permutations and hence is computationally tractable, and derive asymptotic risk bounds for our estimator. In addition, we prove an automatic adaptation property of our estimator for several sub classes of our parameter space which are of natural interest. These sub classes include generalizations of the popular Bradley Terry Model in the Tournament case, the Beta model and Stochastic Block Model in the Graph case, and Holder smooth matrices in the tournament and graph settings.

math.ST↗

arXiv2017

Isotonic regression in general dimensions

We study the least squares regression function estimator over the class of real-valued functions on $[0,1]^d$ that are increasing in each coordinate. For uniformly bounded signals and with a fixed, cubic lattice design, we establish that the estimator achieves the minimax rate of order $n^{-\min\{2/(d+2),1/d\}}$ in the empirical $L_2$ loss, up to poly-logarithmic factors. Further, we prove a sharp oracle inequality, which reveals in particular that when the true regression function is piecewise constant on $k$ hyperrectangles, the least squares estimator enjoys a faster, adaptive rate of convergence of $(k/n)^{\min(1,2/d)}$, again up to poly-logarithmic factors. Previous results are confined to the case $d \leq 2$. Finally, we establish corresponding bounds (which are new even in the case $d=2$) in the more challenging random design setting. There are two surprising features of these results: first, they demonstrate that it is possible for a global empirical risk minimisation procedure to be rate optimal up to poly-logarithmic factors even when the corresponding entropy integral for the function class diverges rapidly; second, they indicate that the adaptation rate for shape-constrained estimators can be strictly worse than the parametric rate.

math.ST↗

arXiv2017

Adaptive Risk Bounds in Unimodal Regression

We study the statistical properties of the least squares estimator in unimodal sequence estimation. Although closely related to isotonic regression, unimodal regression has not been as extensively studied. We show that the unimodal least squares estimator is adaptive in the sense that the risk scales as a function of the number of values in the true underlying sequence. Such adaptivity properties have been shown for isotonic regression by Chatterjee et al(2015) and Bellec(2015). A technical complication in unimodal regression is the non-convexity of the underlying parameter space. We develop a general variational representation of the risk that holds whenever the parameter space can be expressed as a finite union of convex sets, using techniques that may be of interest in other settings.

math.ST↗

arXiv2017

Denoising Flows on Trees

We study the estimation of flows on trees, a structured generalization of isotonic regression. A tree flow is defined recursively as a positive flow value into a node that is partitioned into an outgoing flow to the children nodes, with some amount of the flow possibly leaking outside. We study the behavior of the least squares estimator for flows, and the associated minimax lower bounds. We characterize the risk of the least squares estimator in two regimes. In the first regime the diameter of the tree grows at most logarithmically with the number of nodes. In the second regime, the tree contains long paths. The results are compared with known risk bounds for isotonic regression.

math.ST↗

arXiv2016

Local Minimax Complexity of Stochastic Convex Optimization

We extend the traditional worst-case, minimax analysis of stochastic convex optimization by introducing a localized form of minimax complexity for individual functions. Our main result gives function-specific lower and upper bounds on the number of stochastic subgradient evaluations needed to optimize either the function or its "hardest local alternative" to a given numerical precision. The bounds are expressed in terms of a localized and computational analogue of the modulus of continuity that is central to statistical minimax analysis. We show how the computational modulus of continuity can be explicitly calculated in concrete cases, and relates to the curvature of the function at the optimum. We also prove a superefficiency result that demonstrates it is a meaningful benchmark, acting as a computational analogue of the Fisher information in statistical estimation. The nature and practical implications of the results are demonstrated in simulations.

stat.ML↗

arXiv2016

An Improved Global Risk Bound in Concave Regression

A new risk bound is presented for the problem of convex/concave function estimation, using the least squares estimator. The best known risk bound, as had appeared in \citet{GSvex}, scaled like $\log(en) n^{-4/5}$ under the mean squared error loss, up to a constant factor. The authors in \cite{GSvex} had conjectured that the logarithmic term may be an artifact of their proof. We show that indeed the logarithmic term is unnecessary and prove a risk bound which scales like $n^{-4/5}$ up to constant factors. Our proof technique has one extra peeling step than in a usual chaining type argument. Our risk bound holds in expectation as well as with high probability and also extends to the case of model misspecification, where the true function may not be concave.

math.ST↗

arXiv2015

On matrix estimation under monotonicity constraints

We consider the problem of estimating an unknown $n_1 \times n_2$ matrix $\mathbf{θ^*}$ from noisy observations under the constraint that $\mathbfθ^*$ is nondecreasing in both rows and columns. We consider the least squares estimator (LSE) in this setting and study its risk properties. We show that the worst case risk of the LSE is $n^{-1/2}$, up to multiplicative logarithmic factors, where $n = n_1 n_2$ and that the LSE is minimax rate optimal (up to logarithmic factors). We further prove that for some special $\mathbfθ^*$, the risk of the LSE could be much smaller than $n^{-1/2}$; in fact, it could even be parametric i.e., $n^{-1}$ up to logarithmic factors. Such parametric rates occur when the number of "rectangular" blocks of $\mathbfθ^*$ is bounded from above by a constant. We derive, as a consequence, an interesting adaptation property of the LSE which we term variable adaptation -- the LSE performs as well as the oracle estimator when estimating a matrix that is constant along each row/column. Our proofs borrow ideas from empirical process theory and convex geometry and are of independent interest.

math.ST↗

arXiv2015

On risk bounds in isotonic and other shape restricted regression problems

We consider the problem of estimating an unknown $θ\in {\mathbb{R}}^n$ from noisy observations under the constraint that $θ$ belongs to certain convex polyhedral cones in ${\mathbb{R}}^n$. Under this setting, we prove bounds for the risk of the least squares estimator (LSE). The obtained risk bound behaves differently depending on the true sequence $θ$ which highlights the adaptive behavior of $θ$. As special cases of our general result, we derive risk bounds for the LSE in univariate isotonic and convex regression. We study the risk bound in isotonic regression in greater detail: we show that the isotonic LSE converges at a whole range of rates from $\log n/n$ (when $θ$ is constant) to $n^{-2/3}$ (when $θ$ is uniformly increasing in a certain sense). We argue that the bound presents a benchmark for the risk of any estimator in isotonic regression by proving nonasymptotic local minimax lower bounds. We prove an analogue of our bound for model misspecification where the true $θ$ is not necessarily nondecreasing.

math.ST↗

arXiv2015

Information Theory of Penalized Likelihoods and its Statistical Implications

We extend the correspondence between two-stage coding procedures in data compression and penalized likelihood procedures in statistical estimation. Traditionally, this had required restriction to countable parameter spaces. We show how to extend this correspondence in the uncountable parameter case. Leveraging the description length interpretations of penalized likelihood procedures we devise new techniques to derive adaptive risk bounds of such procedures. We show that the existence of certain countable coverings of the parameter space implies adaptive risk bounds and thus our theory is quite general. We apply our techniques to illustrate risk bounds for $\ell_1$ type penalized procedures in canonical high dimensional statistical problems such as linear regression and Gaussian graphical Models. In the linear regression problem, we also demonstrate how the traditional $l_0$ penalty times $\frac{\log(n)}{2}$ plus lower order terms has a two stage description length interpretation and present risk bounds for this penalized likelihood procedure.

math.ST↗