arXiv subjects

Jeremy Johnson

Publications and source records attributed to Jeremy Johnson.

7 recordsLinked to original sources

arXiv2026

High Rank Matrix Completion via Grassmannian Proxy Fusion

This paper approaches high-rank matrix completion (HRMC) by filling missing entries in a data matrix where columns lie near a union of subspaces, clustering these columns, and identifying the underlying subspaces. Current methods often lack theoretical support, produce uninterpretable results, and require more samples than theoretically necessary. We propose clustering incomplete vectors by grouping proxy subspaces and minimizing two criteria over the Grassmannian: (a) the chordal distance between each point and its corresponding subspace and (b) the geodesic distances between subspaces of all data points. Experiments on synthetic and real datasets demonstrate that our method performs comparably to leading methods in high sampling rates and significantly better in low sampling rates, thus narrowing the gap to the theoretical sampling limit of HRMC.

cs.LG↗

arXiv2023

On Ultrafast X-ray Methods for Magnetism

With the introduction of x-ray free electron laser sources around the world, new scientific approaches for visualizing matter at fundamental length and time-scales have become possible. As it relates to magnetism and "magnetic-type" systems, advanced methods are being developed for studying ultrafast magnetic responses on the time-scales at which they occur. We describe three capabilities which have the potential to seed new directions in this area and present original results from each: pump-probe x-ray scattering with low energy excitation, x-ray photon fluctuation spectroscopy, and ultrafast diffuse x-ray scattering. By combining these experimental techniques with advanced modeling together with machine learning, we describe how the combination of these domains allows for a new understanding in the field of magnetism. Finally, we give an outlook for future areas of investigation and the newly developed instruments which will take us there.

cond-mat.mtrl-sci↗

arXiv2023

TREBUCHET: Fully Homomorphic Encryption Accelerator for Deep Computation

Secure computation is of critical importance to not only the DoD, but across financial institutions, healthcare, and anywhere personally identifiable information (PII) is accessed. Traditional security techniques require data to be decrypted before performing any computation. When processed on untrusted systems the decrypted data is vulnerable to attacks to extract the sensitive information. To address these vulnerabilities Fully Homomorphic Encryption (FHE) keeps the data encrypted during computation and secures the results, even in these untrusted environments. However, FHE requires a significant amount of computation to perform equivalent unencrypted operations. To be useful, FHE must significantly close the computation gap (within 10x) to make encrypted processing practical. To accomplish this ambitious goal the TREBUCHET project is leading research and development in FHE processing hardware to accelerate deep computations on encrypted data, as part of the DARPA MTO Data Privacy for Virtual Environments (DPRIVE) program. We accelerate the major secure standardized FHE schemes (BGV, BFV, CKKS, FHEW, etc.) at >=128-bit security while integrating with the open-source PALISADE and OpenFHE libraries currently used in the DoD and in industry. We utilize a novel tile-based chip design with highly parallel ALUs optimized for vectorized 128b modulo arithmetic. The TREBUCHET coprocessor design provides a highly modular, flexible, and extensible FHE accelerator for easy reconfiguration, deployment, integration and application on other hardware form factors, such as System-on-Chip or alternate chip areas.

cs.CR↗

arXiv2018

Probabilistic Analysis of Block Wiedemann for Leading Invariant Factors

We determine the probability, structure dependent, that the block Wiedemann algorithm correctly computes leading invariant factors. This leads to a tight lower bound for the probability, structure independent. We show, using block size slightly larger than $r$, that the leading $r$ invariant factors are computed correctly with high probability over any field. Moreover, an algorithm is provided to compute the probability bound for a given matrix size and thus to select the block size needed to obtain the desired probability. The worst case probability bound is improved, post hoc, by incorporating the partial information about the invariant factors.

cs.SC↗

arXiv2017

Counting Roots of Polynomials over $\mathbb{Z}/p^2\mathbb{Z}$

Until recently, the only known method of finding the roots of polynomials over prime power rings, other than fields, was brute force. One reason for this is the lack of a division algorithm, obstructing the use of greatest common divisors. Fix a prime $p \in \mathbb{Z}$ and $f \in ( \mathbb{Z}/p^n \mathbb{Z} ) [x]$ any nonzero polynomial of degree $d$ whose coefficients are not all divisible by $p$. For the case $n=2$, we prove a new efficient algorithm to count the roots of $f$ in $\mathbb{Z}/p^2\mathbb{Z}$ within time polynomial in $(d+\operatorname{size}(f)+\log{p})$, and record a concise formula for the number of roots, formulated by Cheng, Gao, Rojas, and Wan.

math.NT↗

arXiv2016

End-to-end learning of brain tissue segmentation from imperfect labeling

Segmenting a structural magnetic resonance imaging (MRI) scan is an important pre-processing step for analytic procedures and subsequent inferences about longitudinal tissue changes. Manual segmentation defines the current gold standard in quality but is prohibitively expensive. Automatic approaches are computationally intensive, incredibly slow at scale, and error prone due to usually involving many potentially faulty intermediate steps. In order to streamline the segmentation, we introduce a deep learning model that is based on volumetric dilated convolutions, subsequently reducing both processing time and errors. Compared to its competitors, the model has a reduced set of parameters and thus is easier to train and much faster to execute. The contrast in performance between the dilated network and its competitors becomes obvious when both are tested on a large dataset of unprocessed human brain volumes. The dilated network consistently outperforms not only another state-of-the-art deep learning approach, the up convolutional network, but also the ground truth on which it was trained. Not only can the incredible speed of our model make large scale analyses much easier but we also believe it has great potential in a clinical setting where, with little to no substantial delay, a patient and provider can go over test results.

cs.CV↗

arXiv2014

Probabilistic analysis of Wiedemann's algorithm for minimal polynomial computation

Blackbox algorithms for linear algebra problems start with projection of the sequence of powers of a matrix to a sequence of vectors (Lanczos), a sequence of scalars (Wiedemann) or a sequence of smaller matrices (block methods). Such algorithms usually depend on the minimal polynomial of the resulting sequence being that of the given matrix. Here exact formulas are given for the probability that this occurs. They are based on the generalized Jordan normal form (direct sum of companion matrices of the elementary divisors) of the matrix. Sharp bounds follow from this for matrices of unknown elementary divisors. The bounds are valid for all finite field sizes and show that a small blocking factor can give high probability of success for all cardinalities and matrix dimensions.

cs.SC↗