arXiv subjects

Jonathan Elmer

Publications and source records attributed to Jonathan Elmer.

At least 19 recordsLinked to original sources

arXiv2026

Learning Under Treatment-Induced Label Indeterminacy with Expert Annotations of Counterfactual Outcomes: A Case Study in Neurological Prognostication

Clinical prediction models are often developed as if the outcome of interest were cleanly observed for every patient. This assumption fails when treatment decisions make the clinically relevant outcome permanently unobservable. As a case study of this problem, we consider post-cardiac-arrest neurological prognostication using a cohort of 2,497 patients, including 1,429 patients whose outcomes were rendered indeterminate by treatment decisions. These patients with indeterminate outcomes were reviewed by independent clinical experts, who provided their guesses of counterfactual outcomes about what would have happened to the patients. We refer to these patients as uncertain cases. We also have patients for whom we observe their clinically relevant outcomes; we refer to these patients as certain cases. We propose a framework for evaluating prediction models that explicitly splits the evaluation between certain and uncertain cases. Here, we cannot easily evaluate both types of cases in a uniform manner as the available target labels differ. We then propose a simple prediction model that uses target labels from both certain and uncertain cases in a manner that allows us to trade off between them. Across the proposed neural model and a collection of tabular baselines, models with similar certain-case AUROC can nevertheless differ substantially in both certain-case Brier score and their probability estimates for uncertain cases. Improving alignment with target labels of uncertain cases for our proposed model generally comes at the cost of worse accuracy on certain cases, highlighting an explicit tradeoff that standard evaluation conceals. These results show that when treatment decisions determine whether clinically meaningful outcomes remain observable, conventional evaluation metrics can miss important failure modes in the very patients for whom prognostic support matters most.

cs.LG↗

arXiv2026

Semi-Invariants of a Matrix and Covector

We prove the following theorem: let $\mathcal{M}_d$ denote the set of $d \times d$ matrices over an infinite field $K$, and let ${(K^d)^*}$ be the set of row vectors. Define an action of $\mathrm{SL}_d(K)$ on $X:= \mathcal{M}_d \oplus (K^d)^*$ by \[ g \cdot (A,\phi) = (gAg^{-1}, \phi g^{-1}).\] Then $K[X]^{\mathrm{SL}_d}$ is a polynomial ring, generated by the coefficients of the characteristic polynomial of $A$ and one further invariant, namely $\Delta(A,\phi):= \det(\phi,\phi A,\phi A^2,\ldots, \phi A^{d-1})^t.$ Our proof is entirely classical in nature, but we give an interpretation of the result and its proof in terms of quiver representation theory.

math.AC↗

arXiv2026

Preventing Data Leakage in EEG-Based Survival Prediction: A Two-Stage Embedding and Transformer Framework

Deep learning models have shown promise in EEG-based outcome prediction for comatose patients after cardiac arrest, but their reliability is often compromised by subtle forms of data leakage. In particular, when long EEG recordings are segmented into short windows and reused across multiple training stages, models may implicitly encode and propagate label information, leading to overly optimistic validation performance and poor generalization. In this study, we identify a previously overlooked form of data leakage in multi-stage EEG modeling pipelines. We demonstrate that violating strict patient-level separation can significantly inflate validation metrics while causing substantial degradation on independent test data. To address this issue, we propose a leakage-aware two-stage framework. In the first stage, short EEG segments are transformed into embedding representations using a convolutional neural network with an ArcFace objective. In the second stage, a Transformer-based model aggregates these embeddings to produce patient-level predictions, with strict isolation between training cohorts to eliminate leakage pathways. Experiments on a large-scale EEG dataset of post-cardiac-arrest patients show that the proposed framework achieves stable and generalizable performance under clinically relevant constraints, particularly in maintaining high sensitivity at stringent specificity thresholds. These results highlight the importance of rigorous data partitioning and provide a practical solution for reliable EEG-based outcome prediction.

cs.LG↗

arXiv2025

The differential invariants of $SL_2(\mathbb{F}_3)$ acting on trace-free matrices over $\mathbb{F}_3$

Let $M$ denote the vector space of $2 \times 2$ matrices with coefficients in $\mathbb{F}_3$ and trace zero. Let $G = SL_2(\mathbb{F}_3)$. Then $G$ acts on $M$ via conjugation. Let $R =(S(M^*) \otimes \Lambda(M^*))$ be the algebra of differential forms on $M$. We compute a minimal generating set for $R^G$ as a commutative-graded algebra. In doing so we utilise the theory of Cohen-Macaulay modules and results in the theory of covariants.

math.AC↗

arXiv2025

STAMP: Spatial-Temporal Adapter with Multi-Head Pooling

Time series foundation models (TSFMs) pretrained on data from multiple domains have shown strong performance on diverse modeling tasks. Various efforts have been made to develop foundation models specific to electroencephalography (EEG) data, which records brain electrical activity as time series. However, no comparative analysis of EEG-specific foundation models (EEGFMs) versus general TSFMs has been performed on EEG-specific tasks. We introduce a novel Spatial-Temporal Adapter with Multi-Head Pooling (STAMP), which leverages univariate embeddings produced by a general TSFM, implicitly models spatial-temporal characteristics of EEG data, and achieves performance comparable to state-of-the-art EEGFMs. A comprehensive analysis is performed on 8 benchmark datasets of clinical tasks using EEG for classification, along with ablation studies. Our proposed adapter is lightweight in trainable parameters and flexible in the inputs it can accommodate, supporting easy modeling of EEG data using TSFMs.

cs.LG↗

arXiv2025

Some formulae relating modular representations of elementary abelian $p$-groups

Let $p>0$ be a prime, $k$ a field of characteristic $p$ and $G$ and elementary abelian $p$-group of order $q = p^n$. Let $W$ be an indecomposable $kG$-module of dimension 2 and define $V_i=S^{i-1}(W^*)$ for each $i=1 \ldots q$. We show that $V_2 \otimes V_i \cong V_{i+1} \oplus V_{i-1}$ provided $i$ is not divisible by $p$, and that $V_2 \otimes V_p$ is indecomposable if $n>1$. Our results generalise results of Almkvist and Fossum for representations of cyclic groups of order $p$. We show how our results give formulae for the direct sum decomposition of $V_i \otimes V_j$ for $i<p$ and $j<j$ modulo summands projective to $\bigoplus_{r=0}^{p-1}V_{rp}$ and conjecture that these formulae extend to the case $i<q$ and $j<q$. We provide some evidence for our conjecture.

math.RT↗

arXiv2025

The separating variety for matrix invariants

Let $G$ be a linear algebraic group defined over an algebraically closed field $k$, and let $V$ be a vector space on which $G$ acts linearly. The separating variety $\mathcal{S}_{G,V}$ is the subvariety of $V^2$ consisting of pairs of points indistinguishable by invariant polynomials in $k[V]^G$. Its geometry places restrictions on the existence of small separating sets, i.e. sets of invariants which distinguish the same points as the full algebra of invariants. The purpose of this article is to study the separating variety in the important special case where $G=\mathrm{GL}_p(\mathbb{C})$ acts on the set $V$ of $n$-tuples of $p \times p$ matrices by simultaneous conjugation. We define a purely combinatorial poset, $\mathcal{P}_{p,n}$, whose maximal elements are in 1-1 correspondence with the irreducible components of $\mathcal{S}_{G,V}$. We show that $\mathcal{S}_{G,V}$ is a variety of dimension $(n+1)p^2-1$, and determine its subdimension for all $n$ and $p$. In particular we show the subdimension is $(n+1)p^2-p$ if $n \geq 3$, or $n \geq 2$ and $p \geq 4$. In the case $n \geq 3$, we give a formula for the number of components of given codimension in $\mathcal{S}_{G,V}$. We give explicit decompositions of $\mathcal{S}_{G,V}$ for all $n$ where $p=2,3$ or $4$. Our results in particular show that when $n\geq 2$ and $p\geq 4$, or $n\geq 3$ and $p=3$, $\mathbb{C}[V]^G$ does not contain a polynomial or hypersurface separating set. It was proven in arXiv:2202.05717 that the same is true if $n \geq 4$ and $p=2$. The author made a conjecture in arXiv:2211.17088 generalising the Skronowski-Weyman theorem for representations of quivers. The results of this paper prove that conjecture in two important special cases: for the quiver with one vertex and an arbitrary number, $n$, of loops, and for the quiver with two vertices and $n$ arrows between them.

math.RT↗

arXiv2025

Stepwise Fine and Gray: Subject-Specific Variable Selection Shows When Hemodynamic Data Improves Prognostication of Comatose Post-Cardiac Arrest Patients

Prognostication for comatose post-cardiac arrest patients is a critical challenge that directly impacts clinical decision-making in the ICU. Clinical information that informs prognostication is collected serially over time. Shortly after cardiac arrest, various time-invariant baseline features are collected (e.g., demographics, cardiac arrest characteristics). After ICU admission, additional features are gathered, including time-varying hemodynamic data (e.g., blood pressure, doses of vasopressor medications). We view these as two phases in which we collect new features. In this study, we propose a novel stepwise dynamic competing risks model that improves the prediction of neurological outcomes by automatically determining when to take advantage of time-invariant features (first phase) and time-varying features (second phase). Notably, our model finds patients for whom this second phase (time-varying hemodynamic) information is beneficial for prognostication and also when this information is beneficial (as we collect more hemodynamic data for a patient over time, how important these data are for prognostication varies). Our approach extends the standard Fine and Gray model to explicitly model the two phases and to incorporate neural networks to flexibly capture complex nonlinear feature relationships. Evaluated on a retrospective cohort of 2,278 comatose post-arrest patients, our model demonstrates robust discriminative performance for the competing outcomes of awakening, withdrawal of life-sustaining therapy, and death despite maximal support. Our approach generalizes to more than two phases in which new features are collected and could be used in other dynamic prediction tasks, where it may be helpful to know when and for whom newly collected features significantly improve prediction.

cs.LG↗

arXiv2025

Cohen-Macaulay modules of covariants for cyclic $p$-groups

Let $G$ be a a finite group, $k$ a field of characteristic dividing $|G|$ and and $V,W$ $kG$-modules. Broer and Chuai showed that if $\mathrm{codim}(V^G) \leq 2$ then the module of covariants $k[V,W]^G = (k[V]\otimes W)^G$ is a Cohen-Macaulay module, hence free over a homogeneous system of parameters for the invariant ring $k[V]^G$. In the present article we prove a general result which allows us to determine whether a set of elements of a free $A$-module is a generating set, for any $k$-algebra $A$. We use this result to find generating sets for all modules of covariants $k[V,W]^G$ over a homogeneous system of parameters, where $\mathrm{codim}(V^G) \leq 2$ and $G$ is a cyclic $p$-group.

math.AC↗

arXiv2025

Perils of Label Indeterminacy: A Case Study on Prediction of Neurological Recovery After Cardiac Arrest

The design of AI systems to assist human decision-making typically requires the availability of labels to train and evaluate supervised models. Frequently, however, these labels are unknown, and different ways of estimating them involve unverifiable assumptions or arbitrary choices. In this work, we introduce the concept of label indeterminacy and derive important implications in high-stakes AI-assisted decision-making. We present an empirical study in a healthcare context, focusing specifically on predicting the recovery of comatose patients after resuscitation from cardiac arrest. Our study shows that label indeterminacy can result in models that perform similarly when evaluated on patients with known labels, but vary drastically in their predictions for patients where labels are unknown. After demonstrating crucial ethical implications of label indeterminacy in this high-stakes context, we discuss takeaways for evaluation, reporting, and design.

cs.LG↗

arXiv2023

Neurological Prognostication of Post-Cardiac-Arrest Coma Patients Using EEG Data: A Dynamic Survival Analysis Framework with Competing Risks

Patients resuscitated from cardiac arrest who enter a coma are at high risk of death. Forecasting neurological outcomes of these patients (the task of neurological prognostication) could help with treatment decisions. In this paper, we propose, to the best of our knowledge, the first dynamic framework for neurological prognostication of post-cardiac-arrest comatose patients using EEG data: our framework makes predictions for a patient over time as more EEG data become available, and different training patients' available EEG time series could vary in length. Predictions are phrased in terms of either time-to-event outcomes (time-to-awakening or time-to-death) or as the patient's probability of awakening or of dying across multiple time horizons. Our framework uses any dynamic survival analysis model that supports competing risks in the form of estimating patient-level cumulative incidence functions. We consider three competing risks as to what happens first to a patient: awakening, being withdrawn from life-sustaining therapies (and thus deterministically dying), or dying (by other causes). We demonstrate our framework by benchmarking three existing dynamic survival analysis models that support competing risks on a real dataset of 922 patients. Our main experimental findings are that: (1) the classical Fine and Gray model which only uses a patient's static features and summary statistics from the patient's latest hour's worth of EEG data is highly competitive, achieving accuracy scores as high as the recently developed Dynamic-DeepHit model that uses substantially more of the patient's EEG data; and (2) in an ablation study, we show that our choice of modeling three competing risks results in a model that is at least as accurate while learning more information than simpler models (using two competing risks or a standard survival analysis setup with no competing risks).

eess.SP↗

arXiv2022

The separating variety for matrix semi-invariants

Let $G$ be a linear algebraic group acting linearly on a vector space $V$, and let $k[V]^G$ be the corresponding algebra of invariant polynomial functions. A separating set $S \subseteq k[V]^G$ is a set of polynomials with the property that for all $v,w \in V$, if there exists $f \in k[V]^G$ separating $v$ and $w$, then there exists $f \in S$ separating $v$ and $w$. In this article we consider the action of $G = \mathrm{SL}_2 \times \mathrm{SL}_2$ on the $\mathbb{C}$-vector space $M_{2,2}^n$ of $n$-tuples of $2 \times 2$ matrices by multiplication on the left and the right. Minimal generating sets $S_n$ of $\mathbb{C}[M_{2,2}^n]^G$ are known, and $|S_n| = \frac{1}{24}(n^4-6n^3+23n^2+6n)$. In recent work, Domokos showed that $S_n$ is a minimal separating set by inclusion, i.e. that no proper subset of $S_n$ is a separating set. Our main result shows that any separating set for $\mathbb{C}[M_{2,2}^n]^G$ has cardinality $\geq 5n-9$. In particular, there is no separating set of size $\dim(\mathbb{C}[M_2^n]^G) = 4n-6$ for $n \geq 4$. We also consider the action of $G= \mathrm{SL}_l(\mathbb{C})$ on $M_{l,n}$ by left multiplication. In that case the algebra of invariants has a minimum generating set of size $\binom{n}{l}$ and dimension $ln-l^2+1$. We show that a separating set for $\mathbb{C}[M_{l,n}]^G$ must have size at least $(2l-2)n-2(l^2-l)$. In particular, $\mathbb{C}[M_{l,n}]^G$ does not contain a separating set of size $\dim(\mathbb{C}[M_{l,n}]^G)$ for $l \geq 3$ and $n \geq l+2$. We include an interpretation of our results in terms of representations of quivers, and make a conjecture generalising the Skowronski-Weyman theorem.

math.AC↗

arXiv2022

The separating variety for 2x2 matrix invariants

Let $G$ be a linear algebraic group acting linearly on a $G$-variety $\mathcal{V}$, and let $k[\mathcal{V}]^G$ be the corresponding algebra of invariant polynomial functions. A separating set $S \subseteq k[\mathcal{V}]^G$ is a set of polynomials with the property that for all $v,w \in \mathcal{V}$, if there exists $f \in k[\mathcal{V}]^G$ separating $v$ and $w$, then there exists $f \in S$ separating $v$ and $w$. In this article we consider the action of $G = \mathrm{GL}_2(\mathbb{C})$ on the variety $\mathcal{M}_2^n$ of $n$-tuples of $2 \times 2$ matrices by simultaneous conjugation. Minimal generating sets $S_n$ of $\mathbb{C}[\mathcal{M}_2^n]^G$ are well-known, and $|S_n| = \frac16(n^3+11n)$. In recent work, Kaygorodov, Lopatin and Popov showed that for all $n \geq 1$, $S_n$ is a minimal separating set by inclusion, i.e. that no proper subset of $S_n$ is a separating set. This does not necessarily mean that $S_n$ has minimum cardinality among all separating sets for $\mathbb{C}[\mathcal{M}_2^n]^G$. Our main result shows that any separating set for $\mathbb{C}[\mathcal{M}_2^n]^G$ has cardinality $\geq 5n-5$. In particular, there is no separating set of size $\dim(\mathbb{C}[\mathcal{M}_2^n]) = 4n-3$ for $n \geq 3$. Further, $S_3$ has indeed minimum cardinality as a separating set, but for $n \geq 4$ there may exist a smaller separating set than $S_n$. We show that for $n \geq 5$ there does, in fact, exist a smaller separating set than $S_n$. We also prove similar results for the left-right action of $\mathrm{SL}_2(\mathbb{C}) \times \mathrm{SL}_2(\mathbb{C})$ on $\mathcal{M}_2^n$.

math.AC↗

arXiv2021

Benefit-aware Early Prediction of Health Outcomes on Multivariate EEG Time Series

Given a cardiac-arrest patient being monitored in the ICU (intensive care unit) for brain activity, how can we predict their health outcomes as early as possible? Early decision-making is critical in many applications, e.g. monitoring patients may assist in early intervention and improved care. On the other hand, early prediction on EEG data poses several challenges: (i) earliness-accuracy trade-off; observing more data often increases accuracy but sacrifices earliness, (ii) large-scale (for training) and streaming (online decision-making) data processing, and (iii) multi-variate (due to multiple electrodes) and multi-length (due to varying length of stay of patients) time series. Motivated by this real-world application, we present BeneFitter that infuses the incurred savings from an early prediction as well as the cost from misclassification into a unified domain-specific target called benefit. Unifying these two quantities allows us to directly estimate a single target (i.e. benefit), and importantly, dictates exactly when to output a prediction: when benefit estimate becomes positive. BeneFitter (a) is efficient and fast, with training time linear in the number of input sequences, and can operate in real-time for decision-making, (b) can handle multi-variate and variable-length time-series, suitable for patient data, and (c) is effective, providing up to 2x time-savings with equal or better accuracy as compared to competitors.

cs.LG↗

arXiv2020

Degree bounds for modular covariants

Let $V,W$ be representations of a cyclic group $G$ of prime order $p$ over a field $k$ of characteristic $p$. The module of covariants $k[V,W]^G$ is the set of $G$-equivariant polynomial maps $V \rightarrow W$, and is a module over $k[V]^G$. We give a formula for the Noether bound $\beta(k[V,W]^G,k[V]^G)$, i.e. the minimal degree $d$ such that $k[V,W]^G$ is generated over $k[V]^G$ by elements of degree at most $d$.

math.AC↗

arXiv2018

On the depth of quotients of modular invariant rings by transfer ideals

Let $G$ be a finite group, and $V$ a finite dimensional vector space over a field $k$ of characteristic dividing the order of $G$. Let $H \leq G$. The transfer map $k[V]^H \rightarrow k[V]^G$ is an important feature of modular invariant theory. Its image is called a transfer ideal $I^G_H$ of $k[V]^G$, and this ideal, along with the quotients $k[V]^G/I^G_H$ are widely studied. In this article we study $k[V]^G/I$, where $I$ is any sum of transfer ideals. Our main result gives an explicit regular sequence of length $\dim(V^G)$ in $k[V]^G/I$ when $G$ is a $p$-group. We identify situations where this is sufficient to compute the depth of $k[V]^G/I$, in particular recovering a result of Totaro. We also study the cases where $G$ is cyclic or isomorphic to the Klein 4 group in greater detail. In particular we use our results to compute the depth of $k[V]^G/I^G_{\{1\}}$ for an arbitrary indecomposable representation of the Klein 4-group.

math.AC↗

arXiv2018

Modular Covariants of Cyclic Groups of Order p

Let $G$ be a cyclic group of order $p$, let $k$ be a field of characteristic $p$, and let $V, W$ be $kG$-modules. We study the modules of covariants $k[V,W]^G = (S(V^*) \otimes W)^G$. For $V$ indecomposable with dimension 2, and $W$ an arbitrary indecomposable module, we show $k[V,W]^G$ is a free $k[V]^G$-module (recovering a result of Broer and Chuai) and we give an explicit set of covariants generating $k[V,W]^G$ freely over $k[V]^G$. For $V$ indecomposable with dimension 3 and $W$ an indecomposable module with dimension at most 5, we show that $k[V,W]^G$ is a Cohen-Macaulay $k[V]^G$-module (again recovering a result of Broer and Chuai) and we give an explicit set of covariants which generate $k[V,W]^G$ freely over a homogeneous system of parameters for $k[V]^G$. We conjecture that a similar set of covariants generates $k[V,W]^G$ freely over a homogeneous system of parameters for $k[V]^G$ when $W$ has arbitrary dimension.

math.AC↗

arXiv2016

Locally finite derivations and modular coinvariants

We consider a finite dimensional $\kk G$-module $V$ of a $p$-group $G$ over a field $\kk$ of characteristic $p$. We describe a generating set for the corresponding Hilbert Ideal. In case $G$ is cyclic this yields that the algebra $\kk[V]_G$ of coinvariants is a free module over its subalgebra generated by $\kk G$-module generators of $V^*$. This subalgebra is a quotient of a polynomial ring by pure powers of its variables. The coinvariant ring was known to have this property only when $G$ was cyclic of prime order, \cite{SezerCoinv}. In addition, we show that if $G$ is the Klein 4-group and $V$ does not contain an indecomposable summand isomorphic to the regular module, then the Hilbert Ideal is a complete intersection, extending a result of the second author and R. J. Shank \cite{SezerShank}.

math.AC↗