Brian Axelrod

Sophie Ostmeier, Brian Axelrod, Jeroen Bertels et al.

Performance metrics for medical image segmentation models are used to measure the agreement between the reference annotation and the predicted segmentation. Usually, overlap metrics, such as the Dice, are used as a metric to evaluate the performance of these models in order for results to be comparable. However, there is a mismatch between the distributions of cases and difficulty level of segmentation tasks in public data sets compared to clinical practice. Common metrics fail to measure the impact of this mismatch, especially for clinical data sets that include low signal pathologies, a difficult segmentation task, and uncertain, small, or empty reference annotations. This limitation may result in ineffective research of machine learning practitioners in designing and optimizing models. Dimensions of evaluating clinical value include consideration of the uncertainty of reference annotations, independence from reference annotation volume size, and evaluation of classification of empty reference annotations. We study how uncertain, small, and empty reference annotations influence the value of metrics for medical image segmentation on an in-house data set regardless of the model. We examine metrics behavior on the predictions of a standard deep learning framework in order to identify metrics with clinical value. We compare to a public benchmark data set (BraTS 2019) with a high-signal pathology and certain, larger, and no empty reference annotations. We may show machine learning practitioners, how uncertain, small, or empty reference annotations require a rethinking of the evaluation and optimizing procedures. The evaluation code was released to encourage further analysis of this topic. https://github.com/SophieOstmeier/UncertainSmallEmpty.git

Sophie Ostmeier, Brian Axelrod, Benjamin F. J. Verhaaren et al.

To determine if a convolutional neural network (CNN) deep learning model can accurately segment acute ischemic changes on non-contrast CT compared to neuroradiologists. Non-contrast CT (NCCT) examinations from 232 acute ischemic stroke patients who were enrolled in the DEFUSE 3 trial were included in this study. Three experienced neuroradiologists independently segmented hypodensity that reflected the ischemic core on each scan. The neuroradiologist with the most experience (expert A) served as the ground truth for deep learning model training. Two additional neuroradiologists (experts B and C) segmentations were used for data testing. The 232 studies were randomly split into training and test sets. The training set was further randomly divided into 5 folds with training and validation sets. A 3-dimensional CNN architecture was trained and optimized to predict the segmentations of expert A from NCCT. The performance of the model was assessed using a set of volume, overlap, and distance metrics using non-inferiority thresholds of 20%, 3ml, and 3mm. The optimized model trained on expert A was compared to test experts B and C. We used a one-sided Wilcoxon signed-rank test to test for the non-inferiority of the model-expert compared to the inter-expert agreement. The final model performance for the ischemic core segmentation task reached a performance of 0.46+-0.09 Surface Dice at Tolerance 5mm and 0.47+-0.13 Dice when trained on expert A. Compared to the two test neuroradiologists the model-expert agreement was non-inferior to the inter-expert agreement, p < 0.05. The CNN accurately delineates the hypodense ischemic core on NCCT in acute ischemic stroke patients with an accuracy comparable to neuroradiologists.

Sophie Ostmeier, Brian Axelrod, Benjamin Pulli et al.

Purpose: Multi-expert deep learning training methods to automatically quantify ischemic brain tissue on Non-Contrast CT Materials and Methods: The data set consisted of 260 Non-Contrast CTs from 233 patients of acute ischemic stroke patients recruited in the DEFUSE 3 trial. A benchmark U-Net was trained on the reference annotations of three experienced neuroradiologists to segment ischemic brain tissue using majority vote and random expert sampling training schemes. We used a one-sided Wilcoxon signed-rank test on a set of segmentation metrics to compare bootstrapped point estimates of the training schemes with the inter-expert agreement and ratio of variance for consistency analysis. We further compare volumes with the 24h-follow-up DWI (final infarct core) in the patient subgroup with full reperfusion and we test volumes for correlation to the clinical outcome (mRS after 30 and 90 days) with the Spearman method. Results: Random expert sampling leads to a model that shows better agreement with experts than experts agree among themselves and better agreement than the agreement between experts and a majority-vote model performance (Surface Dice at Tolerance 5mm improvement of 61% to 0.70 +- 0.03 and Dice improvement of 25% to 0.50 +- 0.04). The model-based predicted volume similarly estimated the final infarct volume and correlated better to the clinical outcome than CT perfusion. Conclusion: A model trained on random expert sampling can identify the presence and location of acute ischemic brain tissue on Non-Contrast CT similar to CT perfusion and with better consistency than experts. This may further secure the selection of patients eligible for endovascular treatment in less specialized hospitals.

Juanwu Lu, Anand Bhaskar, Brian Axelrod et al.

Model merging offers a promising avenue for knowledge integration and parallel development without retraining. Yet, existing methods either ignore the geometry of the loss landscape or rely on intractable full-space Hessian approximations. We propose EpiMer, a framework that casts model merging as solving the Fréchet mean on a Riemannian manifold and restricts the computation to a low-rank subspace spanned by the task vectors. With the expected Hessian as the metric, we reveal a connection between local curvature and epistemic uncertainty of the parameters. Our theoretical analysis decomposes the merging error bound into the subspace Fréchet variance and the residual energy, and provides a closed-form characterization of when curvature-aware merging provably outperforms flat-geometry methods. In addition, our framework unifies both curvature-aware methods and recent spectral methods as special cases of the subspace Fréchet mean with different geometric metrics. Merging fine-tuned CLIP-ViT models on eight image classification tasks, Epistemic Merging strictly outperforms the baselines on all three CLIP-ViT backbones at matched rank, improving the across-task average accuracy and worst-task accuracy on every backbone.

Sophie Ostmeier, Brian Axelrod, Maya Varma et al.

Transformer architectures rely on position encodings to model the spatial structure of input data. Rotary Position Encoding (RoPE) is a widely used method in language models that encodes relative positions through fixed, block-diagonal, rotation matrices applied to key-query interactions. We hypothesize that this inductive bias limits their RoPE's effectiveness for modalities with high dimensional structure. Lie Relative Encodings (LieRE) introduce a principled generalization of RoPE, aimed at increasing the representational capacity of positional encodings in transformers. Instead of fixed 2D rotations, LieRE learns dense skew-symmetric matrices (Lie algebra elements), which are then differentiable mapped to form high-dimensional rotation matrices (Lie group elements). This results in richer, learnable, and continuous, encodings of both relative and absolute positional information. We demonstrate the effectiveness of LieRE on 2D and 3D vision tasks, showing that it generalizes well to higher input resolutions while maintaining computational efficiency. The code and checkpoints are publicly available at https://github.com/StanfordMIMI/LieRE.

Brian Axelrod, Shivam Garg, Yanjun Han et al.

The ``sample amplification'' problem formalizes the following question: Given $n$ i.i.d. samples drawn from an unknown distribution $P$, when is it possible to produce a larger set of $n+m$ samples which cannot be distinguished from $n+m$ i.i.d. samples drawn from $P$? In this work, we provide a firm statistical foundation for this problem by deriving generally applicable amplification procedures, lower bound techniques and connections to existing statistical notions. Our techniques apply to a large class of distributions including the exponential family, and establish a rigorous connection between sample amplification and distribution learning.

Yonadav Shavit, Benjamin Edelman, Brian Axelrod

In many predictive decision-making scenarios, such as credit scoring and academic testing, a decision-maker must construct a model that accounts for agents' propensity to "game" the decision rule by changing their features so as to receive better decisions. Whereas the strategic classification literature has previously assumed that agents' outcomes are not causally affected by their features (and thus that strategic agents' goal is deceiving the decision-maker), we join concurrent work in modeling agents' outcomes as a function of their changeable attributes. As our main contribution, we provide efficient algorithms for learning decision rules that optimize three distinct decision-maker objectives in a realizable linear setting: accurately predicting agents' post-gaming outcomes (prediction risk minimization), incentivizing agents to improve these outcomes (agent outcome maximization), and estimating the coefficients of the true underlying model (parameter estimation). Our algorithms circumvent a hardness result of Miller et al. (2020) by allowing the decision maker to test a sequence of decision rules and observe agents' responses, in effect performing causal interventions through the decision rules.

Brian Axelrod, Shivam Garg, Vatsal Sharan et al.

Given data drawn from an unknown distribution, $D$, to what extent is it possible to ``amplify'' this dataset and output an even larger set of samples that appear to have been drawn from $D$? We formalize this question as follows: an $(n,m)$ $\text{amplification procedure}$ takes as input $n$ independent draws from an unknown distribution $D$, and outputs a set of $m > n$ ``samples''. An amplification procedure is valid if no algorithm can distinguish the set of $m$ samples produced by the amplifier from a set of $m$ independent draws from $D$, with probability greater than $2/3$. Perhaps surprisingly, in many settings, a valid amplification procedure exists, even when the size of the input dataset, $n$, is significantly less than what would be necessary to learn $D$ to non-trivial accuracy. Specifically we consider two fundamental settings: the case where $D$ is an arbitrary discrete distribution supported on $\le k$ elements, and the case where $D$ is a $d$-dimensional Gaussian with unknown mean, and fixed covariance. In the first case, we show that an $\left(n, n + Θ(\frac{n}{\sqrt{k}})\right)$ amplifier exists. In particular, given $n=O(\sqrt{k})$ samples from $D$, one can output a set of $m=n+1$ datapoints, whose total variation distance from the distribution of $m$ i.i.d. draws from $D$ is a small constant, despite the fact that one would need quadratically more data, $n=Θ(k)$, to learn $D$ up to small constant total variation distance. In the Gaussian case, we show that an $\left(n,n+Θ(\frac{n}{\sqrt{d}} )\right)$ amplifier exists, even though learning the distribution to small constant total variation distance requires $Θ(d)$ samples. In both the discrete and Gaussian settings, we show that these results are tight, to constant factors. Beyond these results, we formalize a number of curious directions for future research along this vein.

Brian Axelrod, Leslie Pack Kaelbling, Tomás Lozano-Pérez

As drones and autonomous cars become more widespread it is becoming increasingly important that robots can operate safely under realistic conditions. The noisy information fed into real systems means that robots must use estimates of the environment to plan navigation. Efficiently guaranteeing that the resulting motion plans are safe under these circumstances has proved difficult. We examine how to guarantee that a trajectory or policy is safe with only imperfect observations of the environment. We examine the implications of various mathematical formalisms of safety and arrive at a mathematical notion of safety of a long-term execution, even when conditioned on observational information. We present efficient algorithms that can prove that trajectories or policies are safe with much tighter bounds than in previous work. Notably, the complexity of the environment does not affect our methods ability to evaluate if a trajectory or policy is safe. We then use these safety checking methods to design a safe variant of the RRT planning algorithm.