arXiv subjects

Pranav Singh

Publications and source records attributed to Pranav Singh.

32 records · Page 2Linked to original sources

arXiv2023

Cross-Architectural Positive Pairs improve the effectiveness of Self-Supervised Learning

Existing self-supervised techniques have extreme computational requirements and suffer a substantial drop in performance with a reduction in batch size or pretraining epochs. This paper presents Cross Architectural - Self Supervision (CASS), a novel self-supervised learning approach that leverages Transformer and CNN simultaneously. Compared to the existing state-of-the-art self-supervised learning approaches, we empirically show that CASS-trained CNNs and Transformers across four diverse datasets gained an average of 3.8% with 1% labeled data, 5.9% with 10% labeled data, and 10.13% with 100% labeled data while taking 69% less time. We also show that CASS is much more robust to changes in batch size and training epochs than existing state-of-the-art self-supervised learning approaches. We have open-sourced our code at https://github.com/pranavsinghps1/CASS.

cs.CV↗

arXiv2022

CASS: Cross Architectural Self-Supervision for Medical Image Analysis

Recent advances in deep learning and computer vision have reduced many barriers to automated medical image analysis, allowing algorithms to process label-free images and improve performance. However, existing techniques have extreme computational requirements and drop a lot of performance with a reduction in batch size or training epochs. This paper presents Cross Architectural - Self Supervision (CASS), a novel self-supervised learning approach that leverages Transformer and CNN simultaneously. Compared to the existing state of the art self-supervised learning approaches, we empirically show that CASS-trained CNNs and Transformers across four diverse datasets gained an average of 3.8% with 1% labeled data, 5.9% with 10% labeled data, and 10.13% with 100% labeled data while taking 69% less time. We also show that CASS is much more robust to changes in batch size and training epochs. Notably, one of the test datasets comprised histopathology slides of an autoimmune disease, a condition with minimal data that has been underrepresented in medical imaging. The code is open source and is available on GitHub.

cs.CV↗

arXiv2022

A Data-Efficient Deep Learning Framework for Segmentation and Classification of Histopathology Images

The current study of cell architecture of inflammation in histopathology images commonly performed for diagnosis and research purposes excludes a lot of information available on the biopsy slide. In autoimmune diseases, major outstanding research questions remain regarding which cell types participate in inflammation at the tissue level, and how they interact with each other. While these questions can be partially answered using traditional methods, artificial intelligence approaches for segmentation and classification provide a much more efficient method to understand the architecture of inflammation in autoimmune disease, holding great promise for novel insights. In this paper, we empirically develop deep learning approaches that use dermatomyositis biopsies of human tissue to detect and identify inflammatory cells. Our approach improves classification performance by 26% and segmentation performance by 5%. We also propose a novel post-processing autoencoder architecture that improves segmentation performance by an additional 3%.

eess.IV↗

arXiv2022

Unitarity of some barycentric rational approximants

The exponential function maps the imaginary axis to the unit circle and, for many applications, this unitarity property is also desirable from its approximations. We show that this property is conserved not only by the (k,k)-rational barycentric interpolant of the exponential on the imaginary axis, but also by (k,k)-rational barycentric approximants that minimize a linearized approximation error. These results are a consequence of certain properties of singular vectors of Loewner-type matrices associated to linearized approximation errors. Prominent representatives of this class are rational approximants computed by the adaptive Antoulas--Anderson (AAA) method and the AAA--Lawson method. Our results also lead to a modified procedure with improved numerical stability of the unitarity property and reduced computational cost.

math.NA↗

arXiv2021

Optimal parameters for numerical solvers of PDEs

In this paper we introduce a procedure for identifying optimal methods in parametric families of numerical schemes for initial value problems in partial differential equations. The procedure maximizes accuracy by adaptively computing optimal parameters that minimize a defect-based estimate of the local error at each time-step. Viable refinements are proposed to reduce the computational overheads involved in the solution of the optimization problem, and to maintain conservation properties of the original methods. We apply the new strategy to recently introduced families of conservative schemes for the Korteweg-de Vries equation and for a nonlinear heat equation. Numerical tests demonstrate the improved efficiency of the new technique in comparison with existing methods.

math.NA↗

arXiv2021

Efficient Magnus-type integrators for solar energy conversion in Hubbard models

Strongly interacting electrons in solids are generically described by Hubbardtype models, and the impact of solar light can be modeled by an additional time-dependence. This yields a finite dimensional system of ordinary differential equations (ODE)s of Schrödinger type, which can be solved numerically by exponential time integrators of Magnus type. The efficiency may be enhanced by combining these with operator splittings. We will discuss several different approaches of employing exponential-based methods in conjunction with an adaptive Lanczos method for the evaluation of matrix exponentials and compare their accuracy and efficiency. For each integrator, we use defect-based local error estimators to enable adaptive time-stepping. This serves to reliably control the approximation error and reduce the computational effort

math.NA↗

arXiv2020

Optimal Control of Spins by Analytical Lie Algebraic Derivatives

Computation of derivatives (gradient and Hessian) of a fidelity function is one of the most crucial steps in many optimization algorithms. Having access to accurate methods to calculate these derivatives is even more desired where the optimization process requires propagation of these calculations over many steps, which is in particular important in optimal control of spin systems. Here we propose a novel numerical approach, ESCALADE (Efficient Spin Control using Analytical Lie Algebraic Derivatives) that offers the exact first and second derivatives of the fidelity function by taking advantage of the properties of the Lie group of $2\times 2$ Hermitian matrices, $\mathrm{SU}(2)$, and its Lie algebra, the Lie algebra of skew-Hermitian matrices, $\mathfrak{su}(2)$. A full mathematical treatment of the proposed method along with some numerical examples are presented.

physics.comp-ph↗

arXiv2020

Time adaptive Zassenhaus splittings for the Schrödinger equation in the semiclassical regime

Time dependent Schrödinger equations with conservative force field U commonly constitute a major challenge in the numerical approximation, especially when they are analysed in the semiclassical regime. Extremely high oscillations originate from the semiclassical parameter, and call for appropriate methods. We propose to employ a combination of asymptotic Zassenhaus splitting with time adaptivity. While the former turns the disadvantage of the semiclassical parameter into an advantage, leading to highly efficient methods with low error constants, the latter enables to choose an optimal time step and to speed up the calculations when the oscillations subside. We support the results with numerical examples.

math.NA↗

arXiv2019

Sixth-order schemes for laser--matter interaction in Schrödinger equation

Control of quantum systems via lasers has numerous applications that require fast and accurate numerical solution of the Schrödinger equation. In this paper we present three strategies for extending any sixth-order scheme for Schrödinger equation with time-independent potential to a sixth-order method for Schrödinger equation with laser potential. As demonstrated via numerical examples, these schemes prove effective in the atomic regime as well as the semiclassical regime, and are a particularly appealing alternative to time-ordered exponential splittings when the laser potential is highly oscillatory or known only at specific points in time (on an equispaced grid, for instance). These schemes are derived by exploiting the linear in space form of the time dependent potential under the dipole approximation (whereby commutators in the Magnus expansion reduce to a simpler form), separating the time step of numerical propagation from the issue of adequate time-resolution of the laser field by keeping integrals intact in the Magnus expansion, and eliminating terms with unfavourable structure via carefully designed splittings.

math.NA↗

arXiv2018

Magnus-Zassenhaus methods for the semiclassical Schrödinger equation with oscillatory time-dependent potentials

Schrödinger equations with time-dependent potentials are of central importance in quantum physics and theoretical chemistry, where they aid in the simulation and design of systems and processes at atomic scales. Numerical approximation of these equations is particularly difficult in the semiclassical regime because of the highly oscillatory nature of solution. Highly oscillatory potentials such as lasers compound these difficulties even further. Altogether, these effects render a large number of standard numerical methods less effective in this setting. In this paper we will develop a class of high-order exponential splitting schemes that are able to overcome these challenges by combining the advantages of integral-preserving simplified-commutator Magnus expansions with those of symmetric Zassenhaus splittings. This allows us to use large time steps in our schemes even in the presence of highly oscillatory potentials and solutions.

math.NA↗

arXiv2018

Compact schemes for laser-matter interaction in Schrödinger equation

Numerical solutions for laser-matter interaction in Schrödinger equation has many applications in theoretical chemistry, quantum physics and condensed matter physics. In this paper we introduce a methodology which allows, with a small cost, to extend any fourth-order scheme for Schrödinger equation with time-indepedent potential to a fourth-order method for Schrödinger equation with laser potential. These fourth-order methods improve upon many leading schemes of order six due to their low costs and small error constants.

math.NA↗

arXiv2018

Magnus-Lanczos methods with simplified commutators for the Schrödinger equation with a time-dependent potential

The computation of the Schrödinger equation featuring time-dependent potentials is of great importance in quantum control of atomic and molecular processes. These applications often involve highly oscillatory potentials and require inexpensive but accurate solutions over large spatio-temporal windows. In this work we develop Magnus expansions where commutators have been simplified. Consequently, the exponentiation of these Magnus expansions via Lanczos iterations is significantly cheaper than that for traditional Magnus expansions. At the same time, and unlike most competing methods, we simplify integrals instead of discretising them via quadrature at the outset -- this gives us the flexibility to handle a variety of potentials, being particularly effective in the case of highly oscillatory potentials, where this strategy allows us to consider significantly larger time steps.

math.NA↗

arXiv2016

Efficient methods for time-dependence in semiclassical Schrödinger equations

We build an efficient and unitary (hence stable) method for the solution of the semi-classical Schrödinger equation subject with explicitly time-dependent potentials. The method is based on a combination of the Zassenhaus decomposition (Bader, Iserles, Kropielnicka & Singh 2014) with the Magnus expansion of the time-dependent Hamiltonian. We conclude with numerical experiments.

math.NA↗

arXiv2015

Algebraic theory for higher-order methods in computational quantum mechanics

We present the algebraic foundations of the symmetric Zassenhaus algorithm and some of its variants. These algorithms have proven effective in devising higher-order methods for solving the time-dependent Schrödinger equation in the semiclassical regime. We find that the favourable properties of these methods derive directly from the structural properties of a Z2-graded Lie algebra. Commutators in this Lie algebra can be simplified explicitly, leading to commutator-free methods. Their other structural properties are crucial in proving unitarity, stability, convergence, error bounds and quadratic costs of Zassenhaus based methods. These algebraic structures have also found applications in Magnus expansion based methods for time-varying potentials where they allow significantly milder constraints for convergence and lead to highly effective schemes. The algebraic foundations laid out in this work pave the way for extending higher-order Zassenhaus and Magnus schemes to other equations of quantum mechanics.

math.NA↗