Harm Derksen
YOU?
Author Swipe
View article: Constructive counterexamples to the additivity of minimum output Rényi entropy of quantum channels for all $p>1$
Constructive counterexamples to the additivity of minimum output Rényi entropy of quantum channels for all $p>1$ Open
We present explicit quantum channels with strictly sub-additive minimum output Rényi entropy for all $p>1$, improving upon prior constructions which handled $p>2$. Our example is provided by explicit constructions of linear subspaces with …
View article: Generic orbits, normal bases, and generation degree for fields of rational invariants
Generic orbits, normal bases, and generation degree for fields of rational invariants Open
For a linear representation of a finite group in coprime characteristic, we bound the degrees of the invariant polynomials needed to generate the field of rational invariants as a field, in terms of the degrees of the polynomials needed to…
View article: Estimating the Euclidean distortion of an orbit space
Estimating the Euclidean distortion of an orbit space Open
Given a finite-dimensional inner product space $V$ and a group $G$ of isometries, we consider the problem of embedding the orbit space $V/G$ into a Hilbert space in a way that preserves the quotient metric as well as possible. This inquiry…
View article: Identifying selective PDHK inhibitors using coupled tensor matrix completion and experimental validation
Identifying selective PDHK inhibitors using coupled tensor matrix completion and experimental validation Open
Drug discovery often involves targeting specific members within a family of similar proteins. For example, pyruvate dehydrogenase kinase (PDHK) exists as four isozymes, which exhibit varying expression patterns across multiple tissues. Dif…
View article: X-arability of mixed quantum states
X-arability of mixed quantum states Open
The problem of determining when entanglement is present in a quantum system is one of the most active areas of research in quantum physics. Depending on the setting at hand, different notions of entanglement (or lack thereof) become releva…
View article: General Presentations of Algebras and Foundations of $τ$-tilting Theory
General Presentations of Algebras and Foundations of $τ$-tilting Theory Open
In this short note, we explain how the main results in "$τ$-tilting theory" by Adachi-Iyama-Reiten follow from the results in Section 5 of "General presentations of algebras" by Derksen-Fei.
View article: Boosting uniformity in quasirandom groups: fast and simple
Boosting uniformity in quasirandom groups: fast and simple Open
We study the communication complexity of multiplying $k\times t$ elements from the group $H=\text{SL}(2,q)$ in the number-on-forehead model with $k$ parties. We prove a lower bound of $(t\log H)/c^{k}$. This is an exponential improvement o…
View article: Bi-Lipschitz Quotient embedding for Euclidean Group actions on Data
Bi-Lipschitz Quotient embedding for Euclidean Group actions on Data Open
For the action of the orthogonal group or euclidean group on k-tuples of vectors we construct a bi-Lipschitz embedding from the orbit space into euclidean space.This embedding has distortion sqrt(2).
View article: Continuous sepsis trajectory prediction using tensor-reduced physiological signals
Continuous sepsis trajectory prediction using tensor-reduced physiological signals Open
The quick Sequential Organ Failure Assessment (qSOFA) system identifies an individual's risk to progress to poor sepsis-related outcomes using minimal variables. We used Support Vector Machine, Learning Using Concave and Convex Kernels, an…
View article: Pseudorandomness, symmetry, smoothing: II
Pseudorandomness, symmetry, smoothing: II Open
We prove several new results on the Hamming weight of bounded uniform and small-bias distributions. We exhibit bounded-uniform distributions whose weight is anti-concentrated, matching existing concentration inequalities. This construction…
View article: Subrank and optimal reduction of scalar multiplications to generic tensors
Subrank and optimal reduction of scalar multiplications to generic tensors Open
The subrank of a tensor measures how much a tensor can be diagonalized. We determine this parameter precisely for essentially all (i.e., generic) tensors. Namely, we prove for generic tensors in with that the subrank is . Our result signif…
View article: Pseudorandomness, symmetry, smoothing: I
Pseudorandomness, symmetry, smoothing: I Open
We prove several new results about bounded uniform and small-bias distributions. A main message is that, small-bias, even perturbed with noise, does not fool several classes of tests better than bounded uniformity. We prove this for thresh…
View article: The capacity of quiver representations and the Anantharam-Jog-Nair inequality
The capacity of quiver representations and the Anantharam-Jog-Nair inequality Open
The Anantharam-Jog-Nair inequality [AJN22] in Information Theory provides a unifying approach to the information-theoretic form of the Brascamp-Lieb inequality [CCE09] and the Entropy Power inequality [ZF93]. In this paper, we use methods …
View article: Quadratic Multilinear Discriminant Analysis for Tensorial Data Classification
Quadratic Multilinear Discriminant Analysis for Tensorial Data Classification Open
Over the past decades, there has been an increase of attention to adapting machine learning methods to fully exploit the higher order structure of tensorial data. One problem of great interest is tensor classification, and in particular th…
View article: Tensor Denoising via Amplification and Stable Rank Methods
Tensor Denoising via Amplification and Stable Rank Methods Open
Tensors in the form of multilinear arrays are ubiquitous in data science applications. Captured real-world data, including video, hyperspectral images, and discretized physical systems, naturally occur as tensors and often come with attend…
View article: Polystability in positive characteristic and degree lower bounds for invariant rings
Polystability in positive characteristic and degree lower bounds for invariant rings Open
We develop a representation theoretic technique for detecting closed orbits that is applicable in all characteristics. Our technique is based on Kempf’s theory of optimal subgroups and we make some improvements and simplify the theory from…
View article: A Novel Tropical Geometry-Based Interpretable Machine Learning Method: Pilot Application to Delivery of Advanced Heart Failure Therapies
A Novel Tropical Geometry-Based Interpretable Machine Learning Method: Pilot Application to Delivery of Advanced Heart Failure Therapies Open
—A model’s interpretability is essential to many practical applications such as clinical decision support systems. In this paper, a novel interpretable machine learning method is presented, which can model the relationship between input va…
View article: Quasirandom groups enjoy interleaved mixing
Quasirandom groups enjoy interleaved mixing Open
Let $G$ be a group such that any non-trivial representation has dimension at least $d$. Let $X=(X_{1},X_{2},\ldots,X_{t})$ and $Y=(Y_{1},Y_{2},\ldots,Y_{t})$ be distributions over $G^{t}$. Suppose that $X$ is independent from $Y$. We show …
View article: Maximum likelihood estimation for tensor normal models via castling transforms
Maximum likelihood estimation for tensor normal models via castling transforms Open
In this paper, we study sample size thresholds for maximum likelihood estimation for tensor normal models. Given the model parameters and the number of samples, we determine whether, almost surely, (1) the likelihood function is bounded fr…
View article: Subrank and Optimal Reduction of Scalar Multiplications to Generic Tensors
Subrank and Optimal Reduction of Scalar Multiplications to Generic Tensors Open
Since the seminal works of Strassen and Valiant it has been a central theme in algebraic complexity theory to understand the relative complexity of algebraic problems, that is, to understand which algebraic problems (be it bilinear maps li…
View article: A Novel Tropical Geometry-based Interpretable Machine Learning Method: Application in Prognosis of Advanced Heart Failure
A Novel Tropical Geometry-based Interpretable Machine Learning Method: Application in Prognosis of Advanced Heart Failure Open
A model's interpretability is essential to many practical applications such as clinical decision support systems. In this paper, a novel interpretable machine learning method is presented, which can model the relationship between input var…
View article: Lunar Crater Identification in Digital Images
Lunar Crater Identification in Digital Images Open
It is often necessary to identify a pattern of observed craters in a single image of the lunar surface and without any prior knowledge of the camera’s location. This so-called “lost-in-space” crater identification problem is common in both…
View article: Ranks of linear matrix pencils separate simultaneous similarity orbits
Ranks of linear matrix pencils separate simultaneous similarity orbits Open
This paper solves the two-sided version and provides a counterexample to the general version of the 2003 conjecture by Hadwin and Larson. Consider evaluations of linear matrix pencils $L=T_0+x_1T_1+\cdots+x_mT_m$ on matrix tuples as $L(X_1…
View article: Polystability in positive characteristic and degree lower bounds for invariant rings
Polystability in positive characteristic and degree lower bounds for invariant rings Open
We develop a representation theoretic technique for detecting closed orbits that is applicable in all characteristics. Our technique is based on Kempf's theory of optimal subgroups and we make some improvements and simplify the theory from…
View article: The Capacity of Quiver Representations and Brascamp–Lieb Constants
The Capacity of Quiver Representations and Brascamp–Lieb Constants Open
Let $Q$ be a bipartite quiver, $V$ a real representation of $Q$, and $\sigma $ an integral weight of $Q$ orthogonal to the dimension vector of $V$. Guided by quiver invariant theoretic considerations, we introduce the Brascamp–Lieb (BL) op…
View article: Multimodal tensor-based method for integrative and continuous patient monitoring during postoperative cardiac care
Multimodal tensor-based method for integrative and continuous patient monitoring during postoperative cardiac care Open
Patients recovering from cardiovascular surgeries may develop life-threatening complications such as hemodynamic decompensation, making the monitoring of patients for such complications an essential component of postoperative care. However…
View article: Algebraic Methods for Tensor Data
Algebraic Methods for Tensor Data Open
We develop algebraic methods for computations with tensor data. We give 3 applications: extracting features that are invariant under the orthogonal symmetries in each of the modes, approximation of the tensor spectral norm, and amplificati…
View article: Maximum Likelihood Estimation for Matrix Normal Models via Quiver Representations
Maximum Likelihood Estimation for Matrix Normal Models via Quiver Representations Open
In this paper, we study the log-likelihood function and Maximum Likelihood Estimate (MLE) for the matrix normal model for both real and complex models. We describe the exact number of samples needed to achieve (almost surely) three conditi…