arXiv subjects

Marek Rychlik

Publications and source records attributed to Marek Rychlik.

10 recordsLinked to original sources

arXiv2026

The Evolution of Computer-Assisted Proof In Analysis

The intersection of numerical analysis and machine learning, particularly in the domain of Neural ODEs and Physics-Informed Neural Networks (PINNs), relies heavily on discrete approximations of continuous flows. However, in stiff systems, explicit discretization schemes can induce topological bifurcations, creating spurious attractors that do not exist in the underlying continuous dynamics. In this paper, we analyze a stiff 2D nonlinear system integrated via Heun's method, demonstrating how the discrete map undergoes a numerical bifurcation that renders the true equilibrium repelling along an invariant manifold. Adopting a literate programming paradigm where "the paper is the proof," we embed a Computer-Assisted Proof (CAP) directly within the manuscript. Utilizing rigorous complex interval arithmetic and a dimensionality-reducing "Snap-to-Axis" projection, we mathematically verify that a neighborhood of trajectories is permanently captured by a spurious period-4 sink. This work highlights the critical need for formal safety certifications in learned dynamical models and provides a framework for verifying their stability.

math.DS↗

arXiv2024

On the existence of the maximum likelihood estimate and convergence rate under gradient descent for multi-class logistic regression

We revisit the problem of the existence of the maximum likelihood estimate for multi-class logistic regression. We show that one method of ensuring its existence is by assigning positive probability to every class in the sample dataset. The notion of data separability is not needed, which is in contrast to the classical set up of multi-class logistic regression in which each data sample belongs to one class. We also provide a general and constructive estimate of the convergence rate to the maximum likelihood estimate when gradient descent is used as the optimizer. Our estimate involves bounding the condition number of the Hessian of the maximum likelihood function. The approaches used in this article rely on a simple operator-theoretic framework.

cs.LG↗

arXiv2024

Large-scale data extraction from the UNOS organ donor documents

In this paper we focus on three major task: 1) discussing our methods: Our method captures a portion of the data in DCD flowsheets, kidney perfusion data, and Flowsheet data captured peri-organ recovery surgery. 2) demonstrating the result: We built a comprehensive, analyzable database from 2022 OPTN data. This dataset is by far larger than any previously available even in this preliminary phase; and 3) proving that our methods can be extended to all the past OPTN data and future data. The scope of our study is all Organ Procurement and Transplantation Network (OPTN) data of the USA organ donors since 2008. The data was not analyzable in a large scale in the past because it was captured in PDF documents known as ``Attachments'', whereby every donor's information was recorded into dozens of PDF documents in heterogeneous formats. To make the data analyzable, one needs to convert the content inside these PDFs to an analyzable data format, such as a standard SQL database. In this paper we will focus on 2022 OPTN data, which consists of $\approx 400,000$ PDF documents spanning millions of pages. The entire OPTN data covers 15 years (2008--20022). This paper assumes that readers are familiar with the content of the OPTN data.

cs.CV↗

arXiv2021

A proof of convergence of multi-class logistic regression network

This paper revisits the special type of a neural network known under two names. In the statistics and machine learning community it is known as a multi-class logistic regression neural network. In the neural network community, it is simply the soft-max layer. The importance is underscored by its role in deep learning: as the last layer, whose autput is actually the classification of the input patterns, such as images. Our exposition focuses on mathematically rigorous derivation of the key equation expressing the gradient. The fringe benefit of our approach is a fully vectorized expression, which is a basis of an efficient implementation. The second result of this paper is the positivity of the second derivative of the cross-entropy loss function as function of the weights. This result proves that optimization methods based on convexity may be used to train this network. As a corollary, we demonstrate that no $L^2$-regularizer is needed to guarantee convergence of gradient descent.

stat.ML↗

arXiv2020

Development of a New Image-to-text Conversion System for Pashto, Farsi and Traditional Chinese

We report upon the results of a research and prototype building project \emph{Worldly~OCR} dedicated to developing new, more accurate image-to-text conversion software for several languages and writing systems. These include the cursive scripts Farsi and Pashto, and Latin cursive scripts. We also describe approaches geared towards Traditional Chinese, which is non-cursive, but features an extremely large character set of 65,000 characters. Our methodology is based on Machine Learning, especially Deep Learning, and Data Science, and is directed towards vast quantities of original documents, exceeding a billion pages. The target audience of this paper is a general audience with interest in Digital Humanities or in retrieval of accurate full-text and metadata from digital images.

cs.CV↗

arXiv2019

Deductron -- A Recurrent Neural Network

The current paper is a study in Recurrent Neural Networks (RNN), motivated by the lack of examples simple enough so that they can be thoroughly understood theoretically, but complex enough to be realistic. We constructed an example of structured data, motivated by problems from image-to-text conversion (OCR), which requires long-term memory to decode. Our data is a simple writing system, encoding characters 'X' and 'O' as their upper halves, which is possible due to symmetry of the two characters. The characters can be connected, as in some languages using cursive, such as Arabic (abjad). The string 'XOOXXO' may be encoded as '${\vee}{\wedge}\kern-1.5pt{\wedge}{\vee}\kern-1.5pt{\vee}{\wedge}$'. It follows that we may need to know arbitrarily long past to decode a current character, thus requiring long-term memory. Subsequently we constructed an RNN capable of decoding sequences encoded in this manner. Rather than by training, we constructed our RNN "by inspection", i.e. we guessed its weights. This involved a sequence of steps. We wrote a conventional program which decodes the sequences as the example above. Subsequently, we interpreted the program as a neural network (the only example of this kind known to us). Finally, we generalized this neural network to discover a new RNN architecture whose instance is our handcrafted RNN. It turns out to be a 3 layer network, where the middle layer is capable of performing simple logical inferences; thus the name "deductron". It is demonstrated that it is possible to train our network by simulated annealing. Also, known variants of stochastic gradient descent (SGD) methods are shown to work.

stat.ML↗

arXiv2019

On completion of a linearly independent set to a basis with shifts of a fixed vector

Let $\mathbb{F}$ be an infinite field. Let $n$ be a positive integer and let $1\leq d\leq n$. Let $\vec{f}_1, \vec{f}_2, \ldots, \vec{f}_{d-1} \in \mathbb{F}^{n}$ be $d-1$ linearly independent vectors. Let $\vec{x}=(x_1,x_2,\ldots,x_{d},0,0,\ldots,0)\in\mathbb{F}^{n}$, with $n-d$ zeros at the end. Let $\vec{R}: \mathbb{F}^n \to\mathbb{F}^n$ be the cyclic shift operator to the right, e.g. $\vec{R}\,\vec{x} = (0,x_1,x_2,\ldots,x_{d},0,0,\ldots,0)$. Is there a vector $\vec{x} \in \mathbb{F}^n$, such that the $n-d+1$ vectors $\vec{x},\vec{R}\vec{x}, \ldots ,\vec{R}^{n-d}\vec{x}$ complete the set $\{\vec{f}_j\}_{j=1}^{d-1}$ to a basis of $\mathbb{F}^n$? The answer is in the affirmative for every linearly independent set of $\vec{f}_j$, $j=1,2,\ldots,d-1$. In order to prove this fact, we prove that the $(n-d+1)\times(n-d+1)$ minors of the $(n-d+1)\times(n-d+1)$ circulant matrix. $\begin{bmatrix} \vec{x}, \vec{R} \vec{x}, \ldots, \vec{R}^{n-d} \vec{x} \end{bmatrix}^\intercal$ form a Gröbner basis with respect to the graded reverse lexicographic order (grevlex).

math.RA↗

arXiv2018

Beyond RAID 6 --- an Efficient Systematic Code Protecting Against Multiple Errors, Erasures, and Silent Data Corruption

We describe a replacement for RAID 6, based on a new linear, systematic code, which detects and corrects any combination of $E$ errors (unknown location) and $Z$ erasures (known location) provided that $Z+2E \leq 4$. We investigate some scenarios for error correction beyond the code's minimum distance, using list decoding. We describe a decoding algorithm with quasi-logarithmic time complexity, when parallel processing is used: $\approx O(\log N)$ where $N$ is the number of disks in the array (similar to RAID 6). By comparison, the error correcting code implemented by RAID 6 allows error detection and correction only when $(E,Z)=(1,0)$, $(0,1)$, or $(0,2)$. Hence, when in degraded mode (i.e., when $Z \geq 1$), RAID 6 loses its ability for detecting and correcting random errors (i.e., $E=0$), leading to data loss known as silent data corruption. In contrast, the proposed code does not experience silent data corruption unless $Z \geq 3$. The aforementioned properties of our code, the relative simplicity of implementation, vastly improved data protection, and low computational complexity of the decoding algorithm, make our code a natural successor to RAID 6. As this code is based on the use of quintuple parity, this justifies the name PentaRAID for the RAID technology implementing the ideas of the current paper.

cs.IT↗

arXiv2012

On the Reliability of RAID Systems: An Argument for More Check Drives

In this paper we address issues of reliability of RAID systems. We focus on "big data" systems with a large number of drives and advanced error correction schemes beyond \RAID{6}. Our RAID paradigm is based on Reed-Solomon codes, and thus we assume that the RAID consists of $N$ data drives and $M$ check drives. The RAID fails only if the combined number of failed drives and sector errors exceeds $M$, a property of Reed-Solomon codes. We review a number of models considered in the literature and build upon them to construct models usable for a large number of data and check drives. We attempt to account for a significant number of factors that affect RAID reliability, such as drive replacement or lack thereof, mistakes during service such as replacing the wrong drive, delayed repair, and the finite duration of RAID reconstruction. We evaluate the impact of sector failures that do not result in drive replacement. The reader who needs to consider large $M$ and $N$ will find applicable mathematical techniques concisely summarized here, and should be able to apply them to similar problems. Most methods are based on the theory of continuous time Markov chains, but we move beyond this framework when we consider the fixed time to rebuild broken hard drives, which we model using systems of delay and partial differential equations. One universal statement is applicable across various models: increasing the number of check drives in all cases increases the reliability of the system, and is vastly superior to other approaches of ensuring reliability such as mirroring.

cs.PF↗

arXiv2012

Why is Helfenstein's claim about equichordal points false?

This article explains why a paper by Heinz G. Helfenstein entitled "Ovals with equichordal points", published in J.London Math.Soc.31, 54-57, 1956, is incorrect. We point out a computational error which renders his conclusions invalid. More importantly, we explain that the method cannot be used to solve the equichordal point problem with the method presented there. Today, there is a solution to the problem: Marek R. Rychlik, "A complete solution to the equichordal point problem of Fujiwara, Blaschke, Rothe and Weizenböck", Inventiones Mathematicae 129 (1), 141-212, 1997. However, some mathematicians still point to Helfenstein's paper as a plausible path to a simpler solution. We show that Helfenstein's method cannot be salvaged. The fact that Helfenstein's argument is not correct was known to Wirsing, but he did not explicitly point out the error. This article points out the error and the reasons for the failure of Helfenstein's approach in an accessible, and hopefully enjoyable way.

math.DS↗