Training loss and accuracy are the standard signals used to monitor generalization during deep neural network training. Two well-documented phenomena complicate this picture: in grokking, train loss falls rapidly while test performance improves abruptly only after a long delay; in epoch-wise double descent, train loss decreases monotonically while test loss or error rises and falls. Existing accounts are often task-specific, and a task-agnostic analysis framework for diagnosing and explaining these phenomena across realistic tasks and architectures is missing. We address this challenge by analyzing two competing processes that underlie learning dynamics: representation learning in the encoder and readout calibration in the final classifier. Using tools from representational geometry, neural tangent kernels, and linear probing, we show that both processes are active throughout training, with the fluctuations of their relative speed giving rise to seemingly anomalous generalization dynamics. Applying the representation-readout decomposition to grokking across a wide range of tasks and architectures, we find that the readout is train-biased before grokking onset, and representation learning is gradual but not absent, contrary to the lazy-to-rich account. The framework further provides diagnostic signatures distinguishing spurious from genuine generalization: in a previously reported MNIST grokking example and an epoch-wise double descent example, apparent delayed or non-monotone generalization is shown to arise from representation degradation and readout misalignment induced by non-standard training recipes. Together, these results establish the representation-readout decomposition as a top-down framework for understanding learning dynamics and revealing underlying algorithms for interpretability research.

Generalization, the ability to perform well beyond the training context, is a hallmark of biological and artificial intelligence, yet anticipating unseen failures remains a central challenge. Conventional approaches often take a “bottom-up” mechanistic route by reverse-engineering interpretable features or circuits to build explanatory models. While insightful, these methods often struggle to provide the high-level, predictive signals for anticipating failure in real-world deployment. Here, we propose using a “top-down” approach to studying generalization failures inspired by medical biomarkers: identifying system-level measurements that serve as robust indicators of a model's future performance. Rather than mapping out detailed internal mechanisms, we systematically design and test network markers to probe structure, function links, identify prognostic indicators, and validate predictions in real-world settings. In image classification, we find that task-relevant geometric properties of in-distribution (ID) object manifolds consistently forecast poor out-of-distribution (OOD) generalization. In particular, reductions in two geometric measures, effective manifold dimensionality and utility, predict weaker OOD performance across diverse architectures, optimizers, and datasets. We apply this finding to transfer learning with ImageNet-pretrained models. We consistently find that the same geometric patterns predict OOD transfer performance more reliably than ID accuracy. This work demonstrates that representational geometry can expose hidden vulnerabilities, offering more robust guidance for model selection and AI interpretability.

Brains and artificial neural networks compute with continuous variables such as object position or stimulus orientation. However, the complex variability in neural responses makes it difficult to link internal representational structure to task performance. We develop a statistical-mechanical theory of regression capacity that relates linear decoding efficiency of continuous variables to geometric properties of neural manifolds. Our theory handles complex neural variability and applies to real data, revealing increasing capacity for decoding object position and size along the monkey visual stream.

Integrating task-relevant information into neural representations is a fundamental ability of both biological and artificial intelligence systems. Recent theories have categorized learning into two regimes: the rich regime, where neural networks actively learn task-relevant features, and the lazy regime, where networks behave like random feature models. Yet this simple lazy-rich dichotomy overlooks a diverse underlying taxonomy of feature learning, shaped by differences in learning algorithms, network architectures, and data properties. To address this gap, we introduce an analysis framework to study feature learning via the geometry of neural representations. Rather than inspecting individual learned features, we characterize how task-relevant representational manifolds evolve throughout the learning process. We show, in both theoretical and empirical settings, that as networks learn features, task-relevant manifolds untangle, with changes in manifold geometry revealing distinct learning stages and strategies beyond the lazy-rich dichotomy. This framework provides novel insights into feature learning across neuroscience and machine learning, shedding light on structural inductive biases in neural circuits and the mechanisms underlying out-of-distribution generalization.

Understanding how neural systems efficiently process information through distributed representations is a fundamental challenge at the interface of neuroscience and machine learning. Recent approaches analyze the statistical and geometrical attributes of neural representations as population-level mechanistic descriptors of task implementation. In particular, manifold capacity has emerged as a promising framework linking population geometry to the separability of neural manifolds. However, this metric has been limited to linear readouts. To address this limitation, we introduce a theoretical framework that leverages latent directions in input space, which can be related to contextual information. We derive an exact formula for the context-dependent manifold capacity that depends on manifold geometry and context correlations, and validate it on synthetic and real data. Our framework's increased expressivity captures representation reformatting in deep networks at early stages of the layer hierarchy, previously inaccessible to analysis. As context-dependent nonlinearity is ubiquitous in neural systems, our data-driven and theoretically grounded approach promises to elucidate context-dependent computation across scales, datasets, and models.

Cognitive brain functions rely on experience-dependent internal representations of relevant information. Such representations are organized by attractor dynamics or other mechanisms that constrain population activity onto "neural manifolds". Quantitative analyses of representational manifolds are complicated by their potentially complex geometry, particularly in the absence of attractor states. Here we trained juvenile and adult zebrafish in an odor discrimination task and measured neuronal population activity to analyze representations of behaviorally relevant odors in telencephalic area pDp, the homolog of piriform cortex. No obvious signatures of attractor dynamics were detected. However, olfactory discrimination training selectively enhanced the separation of neural manifolds representing task-relevant odors from other representations, consistent with predictions of autoassociative network models endowed with precise synaptic balance. Analytical approaches using the framework of manifold capacity revealed multiple geometrical modifications of representational manifolds that supported the classification of task-relevant sensory information. Manifold capacity predicted odor discrimination across individuals better than other descriptors of population activity, indicating a close link between manifold geometry and behavior. Hence, pDp and possibly related recurrent networks store information in the geometry of representational manifolds, resulting in joint sensory and semantic maps that may support distributed learning processes.

Decoder-only language models have the ability to dynamically switch between various computational tasks based on input prompts. Despite many successful applications of prompting, there is very limited understanding of the internal mechanism behind such flexibility. In this work, we investigate how different prompting methods affect the geometry of representations in these models. Employing a framework grounded in statistical physics, we reveal that various prompting techniques, while achieving similar performance, operate through distinct representational mechanisms for task adaptation. Our analysis highlights the critical role of input distribution samples and label semantics in few-shot in-context learning. We also demonstrate evidence of synergistic and interfering interactions between different tasks on the representational level. Our work contributes to the theoretical understanding of large language models and lays the groundwork for developing more effective, representation-aware prompting strategies.

From an eagle spotting a fish in shimmering water to a scientist extracting patterns from noisy data, many cognitive tasks require untangling overlapping signals. Neural circuits achieve this by transforming complex sensory inputs into distinct, separable representations that guide behavior. Data-visualization techniques convey the geometry of these transformations, and decoding approaches quantify performance efficiency. However, we lack a framework for linking these two key aspects. Here we address this gap by introducing a data-driven analysis framework, which we call Geometry Linked to Untangling Efficiency (GLUE) with manifold capacity theory, that links changes in the geometrical properties of neural activity patterns to representational untangling at the computational level. We applied GLUE to over seven neuroscience datasets-spanning multiple organisms, tasks, and recording techniques-and found that task-relevant representations untangle in many domains, including along the cortical hierarchy, through learning, and over the course of intrinsic neural dynamics. Furthermore, GLUE can characterize the underlying geometric mechanisms of representational untangling, and explain how it facilitates efficient and robust computation. Beyond neuroscience, GLUE provides a powerful framework for quantifying information organization in data-intensive fields such as structural genomics and interpretable AI, where analyzing high-dimensional representations remains a fundamental challenge.

The study of the brain encompasses multiple scales, including temporal, spatial, and functional aspects. To integrate understanding across these different levels and modalities, it requires developing quantification methods and frameworks. Here, we present effective Geometric measures from Correlated Manifold Capacity theory (GCMC) for probing the functional structure in neural representations. We utilize a statistical physics approach to establish analytical connections between neural co-variabilities and downstream read-out efficiency. These effective geometric measures capture both stimulus-driven and behavior-driven structures in neural population activities, while extracting computationally-relevant information from neural data into intuitive and interpretable analysis descriptors. We apply GCMC to a diverse collection of datasets with different recording methods, various model organisms, and multiple task modalities. Specifically, we demonstrate that GCMC enables a wide range of multi-scale data analysis. This includes quantifying the spatial progression of encoding efficiency across brain regions, revealing the temporal dynamics of task-relevant manifold geometry in information processing, and characterizing variances as well as invariances in neural representations throughout learning. Lastly, the effective manifold geometric measures may be viewed as order parameters for phases related to computational efficiency, facilitating data-driven hypothesis generation and latent embedding.### Competing Interest StatementThe authors have declared no competing interest.

Recently, growth in our understanding of the computations performed in both biological and artificial neural networks has largely been driven by either low-level mechanistic studies or global normative approaches. However, concrete methodologies for bridging the gap between these levels of abstraction remain elusive. In this work, we investigate the internal mechanisms of neural networks through the lens of neural population geometry, aiming to provide understanding at an intermediate level of abstraction, as a way to bridge that gap. Utilizing manifold capacity theory (MCT) from statistical physics and manifold alignment analysis (MAA) from high-dimensional statistics, we probe the underlying organization of task-dependent manifolds in deep neural networks and macaque neural recordings. Specifically, we quantitatively characterize how different learning objectives lead to differences in the organizational strategies of these models and demonstrate how these geometric analyses are connected to the decodability of task-relevant information. These analyses present a strong direction for bridging mechanistic and normative theories in neural networks through neural population geometry, potentially opening up many future research avenues in both machine learning and neuroscience.

Demonstrating quantum advantage requires experimental implementation of a computational task that is hard to achieve using state-of-the-art classical systems. One approach is to perform sampling from a probability distribution associated with a class of highly entangled many-body wavefunctions. It has been suggested that this approach can be certified with the Linear Cross-Entropy Benchmark (XEB). We critically examine this notion. First, in a"benign"setting where an honest implementation of noisy quantum circuits is assumed, we characterize the conditions under which the XEB approximates the fidelity. Second, in an"adversarial"setting where all possible classical algorithms are considered for comparison, we show that achieving relatively high XEB values does not imply faithful simulation of quantum dynamics. We present an efficient classical algorithm that, with 1 GPU within 2s, yields high XEB values, namely 2-12% of those obtained in experiments. By identifying and exploiting several vulnerabilities of the XEB, we achieve high XEB values without full simulation of quantum circuits. Remarkably, our algorithm features better scaling with the system size than noisy quantum devices for commonly studied random circuit ensembles. To quantitatively explain the success of our algorithm and the limitations of the XEB, we use a theoretical framework in which the average XEB and fidelity are mapped to statistical models. We illustrate the relation between the XEB and the fidelity for quantum circuits in various architectures, with different gate choices, and in the presence of noise. Our results show that XEB's utility as a proxy for fidelity hinges on several conditions, which must be checked in the benign setting but cannot be assumed in the adversarial setting. Thus, the XEB alone has limited utility as a benchmark for quantum advantage. We discuss ways to overcome these limitations.

A constraint satisfaction problem (CSP), \(\textsf {Max-CSP}(\mathcal {F})\) , is specified by a finite set of constraints \(\mathcal {F}\subseteq \lbrace [q]^k \rightarrow \lbrace 0,1\rbrace \rbrace\) for positive integers q and k . An instance of the problem on n variables is given by m applications of constraints from \(\mathcal {F}\) to subsequences of the n variables, and the goal is to find an assignment to the variables that satisfies the maximum number of constraints. In the (γ ,β)-approximation version of the problem for parameters 0 ≤ β ≤ γ ≤ 1, the goal is to distinguish instances where at least γ fraction of the constraints can be satisfied from instances where at most β fraction of the constraints can be satisfied. In this work, we consider the approximability of this problem in the context of sketching algorithms and give a dichotomy result. Specifically, for every family \(\mathcal {F}\) and every β < γ, we show that either a linear sketching algorithm solves the problem in polylogarithmic space or the problem is not solvable by any sketching algorithm in \(o(\sqrt {n})\) space. In particular, we give non-trivial approximation algorithms using polylogarithmic space for infinitely many constraint satisfaction problems. We also extend previously known lower bounds for general streaming algorithms to a wide variety of problems, and in particular the case of q = k =2, where we get a dichotomy, and the case when the satisfying assignments of the constraints of \(\mathcal {F}\) support a distribution on \([q]^k\) with uniform marginals. Prior to this work, other than sporadic examples, the only systematic classes of CSPs that were analyzed considered the setting of Boolean variables q = 2, binary constraints k =2, and singleton families \(|\mathcal {F}|=1\) and only considered the setting where constraints are placed on literals rather than variables. Our positive results show wide applicability of bias-based algorithms used previously by [ 47 ] and [ 41 ], which we extend to include richer norm estimation algorithms, by giving a systematic way to discover biases. Our negative results combine the Fourier analytic methods of [ 56 ], which we extend to a wider class of CSPs, with a rich collection of reductions among communication complexity problems that lie at the heart of the negative results. In particular, previous works used Fourier analysis over the Boolean cube to initiate their results and the results seemed particularly tailored to functions on Boolean literals (i.e., with negations). Our techniques surprisingly allow us to get to general q -ary CSPs without negations by appealing to the same Fourier analytic starting point over Boolean hypercubes.

Higher brain functions depend on experience-dependent representations of relevant information that may be organized by attractor dynamics or by geometrical modifications of continuous "neural manifolds". To explore these scenarios we analyzed odor-evoked activity in telencephalic area pDp of juvenile and adult zebrafish, the homolog of piriform cortex. No obvious signatures of attractor dynamics were detected. Rather, olfactory discrimination training selectively enhanced the separation of neural manifolds representing task-relevant odors from other representations, consistent with predictions of autoassociative network models endowed with precise synaptic balance. Analytical approaches using the framework of manifold capacity revealed multiple geometrical modifications of representational manifolds that supported the classification of task-relevant sensory information. Manifold capacity predicted odor discrimination across individuals, indicating a close link between manifold geometry and behavior. Hence, pDp and possibly related recurrent networks store information in the geometry of representational manifolds, resulting in joint sensory and semantic maps that may support distributed learning processes.

Two transformative waves of computing have redefined the way we approach science. The first wave came with the birth of the digital computer, which enabled scientists to numerically simulate their models and analyze massive datasets. This technological breakthrough led to the emergence of many sub-disciplines bearing the prefix "computational" in their names. Currently, we are in the midst of the second wave, marked by the remarkable advancements in artificial intelligence. From predicting protein structures to classifying galaxies, the scope of its applications is vast, and there can only be more awaiting us on the horizon. While these two waves influence scientific methodology at the instrumental level, in this dissertation, I will present the computational lens in science, aiming at the conceptual level. Specifically, the central thesis posits that computation serves as a convenient and mechanistic language for understanding and analyzing information processing systems, offering the advantages of composability and modularity. This dissertation begins with an illustration of the blueprint of the computational lens, supported by a review of relevant previous work. Subsequently, I will present my own works in quantum physics and neuroscience as concrete examples. In the concluding chapter, I will contemplate the potential of applying the computational lens across various scientific fields, in a way that can provide significant domain insights, and discuss potential future directions.

Social learning (SL) through experience with conspecifics can facilitate the acquisition of many behaviors. Thus, when Mongolian gerbils are exposed to a demonstrator performing an auditory discrimination task, their subsequent task acquisition is facilitated, even in the absence of visual cues. Here, we show that transient inactivation of auditory cortex (AC) during exposure caused a significant delay in task acquisition during the subsequent practice phase, suggesting that AC activity is necessary for SL. Moreover, social exposure induced an improvement in AC neuron sensitivity to auditory task cues. The magnitude of neural change during exposure correlated with task acquisition during practice. In contrast, exposure to only auditory task cues led to poorer neurometric and behavioral outcomes. Finally, social information during exposure was encoded in the AC of observer animals. Together, our results suggest that auditory SL is supported by AC neuron plasticity occurring during social exposure and prior to behavioral performance.

Neural networks are known to use spurious correlations such as background information for classification. While prior work has looked at spurious correlations that are widespread in the training data, in this work, we investigate how sensitive neural networks are to rare spurious correlations, which may be harder to detect and correct, and may lead to privacy leaks. We introduce spurious patterns correlated with a fixed class to a few training examples and find that it takes only a handful of such examples for the network to learn the correlation. Furthermore, these rare spurious correlations also impact accuracy and privacy. We empirically and theoretically analyze different factors involved in rare spurious correlations and propose mitigation methods accordingly. Specifically, we observe that $\ell_2$ regularization and adding Gaussian noise to inputs can reduce the undesirable effects. Code available at https://github.com/yangarbiter/rare-spurious-correlation.

We analyze the sketching approximability of constraint satisfaction problems on Boolean domains, where the constraints are balanced linear threshold functions applied to literals. In~particular, we explore the approximability of monarchy-like functions where the value of the function is determined by a weighted combination of the vote of the first variable (the president) and the sum of the votes of all remaining variables. The pure version of this function is when the president can only be overruled by when all remaining variables agree. For every $k \geq 5$, we show that CSPs where the underlying predicate is a pure monarchy function on $k$ variables have no non-trivial sketching approximation algorithm in $o(\sqrt{n})$ space. We also show infinitely many weaker monarchy functions for which CSPs using such constraints are non-trivially approximable by $O(\log(n))$ space sketching algorithms. Moreover, we give the first example of sketching approximable asymmetric Boolean CSPs. Our results work within the framework of Chou, Golovnev, Sudan, and Velusamy (FOCS 2021) that characterizes the sketching approximability of all CSPs. Their framework can be applied naturally to get a computer-aided analysis of the approximability of any specific constraint satisfaction problem. The novelty of our work is in using their work to get an analysis that applies to infinitely many problems simultaneously.

We consider the approximability of constraint satisfaction problems in the streaming setting. For every constraint satisfaction problem (CSP) on n variables taking values in {0,…,q−1}, we prove that improving over the trivial approximability by a factor of q requires Ω(n) space even on instances with O(n) constraints. We also identify a broad subclass of problems for which any improvement over the trivial approximability requires Ω(n) space. The key technical core is an optimal, q−(k−1)-inapproximability for the Max k-LIN-mod q problem, which is the Max CSP problem where every constraint is given by a system of k−1 linear equations mod q over k variables. Our work builds on and extends the breakthrough work of Kapralov and Krachun (Proc. STOC 2019) who showed a linear lower bound on any non-trivial approximation of the MaxCut problem in graphs. MaxCut corresponds roughly to the case of Max k-LIN-mod q with k=q=2. For general CSPs in the streaming setting, prior results only yielded Ω(√n) space bounds. In particular no linear space lower bound was known for an approximation factor less than 1/2 for any CSP. Extending the work of Kapralov and Krachun to Max k-LIN-mod q to k>2 and q>2 (while getting optimal hardness results) is the main technical contribution of this work. Each one of these extensions provides non-trivial technical challenges that we overcome in this work.

Neuromorphic computing would benefit from the utilization of improved customized hardware. However, the translation of neuromorphic algorithms to hardware is not easily accomplished. In particular, building superconducting neuromorphic systems requires expertise in both superconducting physics and theoretical neuroscience, which makes such design particularly challenging. In this work, we aim to bridge this gap by presenting a tool and methodology to translate algorithmic parameters into circuit specifications. We first show the correspondence between theoretical neuroscience models and the dynamics of our circuit topologies. We then apply this tool to solve a linear system and implement Boolean logic gates by creating spiking neural networks with our superconducting nanowire-based hardware.

A constraint satisfaction problem (CSP), $\text{Max}-\text{CSP}(\mathcal{F})$, is specified by a finite set of constraints $\mathcal{F}\subseteq\{[q]^{k}\rightarrow\{0,1\}\}$ for positive integers $q$ and $k$. An instance of the problem on $n$ variables is given by $m$ applications of constraints from $\mathcal{F}$ to subsequences of the $n$ variables, and the goal is to find an assignment to the variables that satisfies the maximum number of constraints. In the ($\gamma, \beta$)-approximation version of the problem, for parameters $0\leq\beta < \gamma\leq 1$, the goal is to distinguish instances where at least $\gamma$ fraction of the constraints can be satisfied from instances where at most $\beta$ fraction of the constraints can be satisfied. In this work we consider the approximability of this problem in the context of sketching algorithms and give a dichotomy result. Specifically, for every family $\mathcal{F}$ and every $\beta < \gamma$, we show that either a linear sketching algorithm solves the problem in polylogarithmic space, or the problem is not solvable by any sketching algorithm in $o(\sqrt{n})$ space. We also extend previously known lower bounds for general streaming algorithms to a wide variety of problems, and in particular the case of $q=k=2$ where we get a dichotomy and the case when the satisfying assignments of $f$ support a distribution on $[q]^{k}$ with uniform marginals. Prior to this work, other than sporadic examples, the only systematic class of CSPs that were analyzed considered the setting of Boolean variables $q=2$, binary constraints $k=2$, singleton families $\vert \mathcal{F}\vert =1$ and only considered the setting where constraints are placed on literals rather than variables. Our positive results show wide applicability of bias-based algorithms used previously by [2] and [3], which we extend to include richer norm estimation algorithms, by giving a systematic way to discover biases. Our negative results combine the Fourier analytic methods of [4], which we extend to a wider class of CSPs, with a rich collection of reductions among communication complexity problems that lie at the heart of the negative results. In particular, previous works used Fourier analysis over the Boolean cube to initiate their results and the results seemed particularly tailored to functions on Boolean literals (i.e., with negations). Our techniques surprisingly allow us to get to general $q$-ary CSPs without negations by appealing to the same Fourier analytic starting point over Boolean hypercubes.