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The optimal training of a vision transformer for brain encoding depends on three factors: model size, data size, and computational resources. This study investigates these three pillars, focusing on the effects of data scaling, model scaling, and high-performance computing on brain encoding results. Using VideoGPT to extract efficient spatiotemporal features from videos and training a Ridge model to predict brain activity based on these features, we conducted benchmark experiments with varying data sizes (10k, 100k, 1M, 6M) and different model configurations of GPT-2, including hidden layer dimensions, number of layers, and number of attention heads. We also evaluated the effects of training models with 32-bit vs 16-bit floating point representations. Our results demonstrate that increasing the hidden layer dimensions significantly improves brain encoding performance, as evidenced by higher Pearson correlation coefficients across all subjects. In contrast, the number of attention heads does not have a significant effect on the encoding results. Additionally, increasing the number of layers shows some improvement in brain encoding correlations, but the trend is not as consistent as that observed with hidden layer dimensions. The data scaling results show that larger training datasets lead to improved brain encoding performance, with the highest Pearson correlation coefficients observed for the largest dataset size (6M). These findings highlight that the effects of data scaling are more significant compared to model scaling in enhancing brain encoding performance. Furthermore, we explored the impact of floating-point precision by comparing 32-bit and 16-bit representations. Training with 16-bit precision yielded the same brain encoding accuracy as 32-bit, while reducing training time by 1.17 times, demonstrating its efficiency for high-performance computing tasks.

This paper studies the generalization performance of iterates obtained by Gradient Descent (GD), Stochastic Gradient Descent (SGD) and their proximal variants in high-dimensional robust regression problems. The number of features is comparable to the sample size and errors may be heavy-tailed. We introduce estimators that precisely track the generalization error of the iterates along the trajectory of the iterative algorithm. These estimators are provably consistent under suitable conditions. The results are illustrated through several examples, including Huber regression, pseudo-Huber regression, and their penalized variants with non-smooth regularizer. We provide explicit generalization error estimates for iterates generated from GD and SGD, or from proximal SGD in the presence of a non-smooth regularizer. The proposed risk estimates serve as effective proxies for the actual generalization error, allowing us to determine the optimal stopping iteration that minimizes the generalization error. Extensive simulations confirm the effectiveness of the proposed generalization error estimates., Comment: Camera-ready version of NeurIPS 2024 paper

We characterize the squared prediction risk of ensemble estimators obtained through subagging (subsample bootstrap aggregating) regularized M-estimators and construct a consistent estimator for the risk. Specifically, we consider a heterogeneous collection of $M \ge 1$ regularized M-estimators, each trained with (possibly different) subsample sizes, convex differentiable losses, and convex regularizers. We operate under the proportional asymptotics regime, where the sample size $n$, feature size $p$, and subsample sizes $k_m$ for $m \in [M]$ all diverge with fixed limiting ratios $n/p$ and $k_m/n$. Key to our analysis is a new result on the joint asymptotic behavior of correlations between the estimator and residual errors on overlapping subsamples, governed through a (provably) contractible nonlinear system of equations. Of independent interest, we also establish convergence of trace functionals related to degrees of freedom in the non-ensemble setting (with $M = 1$) along the way, extending previously known cases for square loss and ridge, lasso regularizers. When specialized to homogeneous ensembles trained with a common loss, regularizer, and subsample size, the risk characterization sheds some light on the implicit regularization effect due to the ensemble and subsample sizes $(M,k)$. For any ensemble size $M$, optimally tuning subsample size yields sample-wise monotonic risk. For the full-ensemble estimator (when $M \to \infty$), the optimal subsample size $k^\star$ tends to be in the overparameterized regime $(k^\star \le \min\{n,p\})$, when explicit regularization is vanishing. Finally, joint optimization of subsample size, ensemble size, and regularization can significantly outperform regularizer optimization alone on the full data (without any subagging)., Comment: 72 pages, 14 figures, 6 tables

This paper investigates the iterates $\hbb^1,\dots,\hbb^T$ obtained from iterative algorithms in high-dimensional linear regression problems, in the regime where the feature dimension $p$ is comparable with the sample size $n$, i.e., $p \asymp n$. The analysis and proposed estimators are applicable to Gradient Descent (GD), proximal GD and their accelerated variants such as Fast Iterative Soft-Thresholding (FISTA). The paper proposes novel estimators for the generalization error of the iterate $\hbb^t$ for any fixed iteration $t$ along the trajectory. These estimators are proved to be $\sqrt n$-consistent under Gaussian designs. Applications to early-stopping are provided: when the generalization error of the iterates is a U-shape function of the iteration $t$, the estimates allow to select from the data an iteration $\hat t$ that achieves the smallest generalization error along the trajectory. Additionally, we provide a technique for developing debiasing corrections and valid confidence intervals for the components of the true coefficient vector from the iterate $\hbb^t$ at any finite iteration $t$. Extensive simulations on synthetic data illustrate the theoretical results.

This paper studies the asymptotics of resampling without replacement in the proportional regime where dimension $p$ and sample size $n$ are of the same order. For a given dataset $(X,y)\in \mathbb{R}^{n\times p}\times \mathbb{R}^n$ and fixed subsample ratio $q\in(0,1)$, the practitioner samples independently of $(X,y)$ iid subsets $I_1,...,I_M$ of $\{1,...,n\}$ of size $q n$ and trains estimators $\hat{\beta}(I_1),...,\hat{\beta}(I_M)$ on the corresponding subsets of rows of $(X, y)$. Understanding the performance of the bagged estimate $\bar{\beta} = \frac1M\sum_{m=1}^M \hat{\beta}(I_1),...,\hat{\beta}(I_M)$, for instance its squared error, requires us to understand correlations between two distinct $\hat{\beta}(I_m)$ and $\hat{\beta}(I_{m'})$ trained on different subsets $I_m$ and $I_{m'}$. In robust linear regression and logistic regression, we characterize the limit in probability of the correlation between two estimates trained on different subsets of the data. The limit is characterized as the unique solution of a simple nonlinear equation. We further provide data-driven estimators that are consistent for estimating this limit. These estimators of the limiting correlation allow us to estimate the squared error of the bagged estimate $\bar{\beta}$, and for instance perform parameter tuning to choose the optimal subsample ratio $q$. As a by-product of the proof argument, we obtain the limiting distribution of the bivariate pair $(x_i^T \hat{\beta}(I_m), x_i^T \hat{\beta}(I_{m'}))$ for observations $i\in I_m\cap I_{m'}$, i.e., for observations used to train both estimates., Comment: 25 pages, 10 figures

Brain encoding with neuroimaging data is an established analysis aimed at predicting human brain activity directly from complex stimuli features such as movie frames. Typically, these features are the latent space representation from an artificial neural network, and the stimuli are image, audio, or text inputs. Ridge regression is a popular prediction model for brain encoding due to its good out-of-sample generalization performance. However, training a ridge regression model can be highly time-consuming when dealing with large-scale deep functional magnetic resonance imaging (fMRI) datasets that include many space-time samples of brain activity. This paper evaluates different parallelization techniques to reduce the training time of brain encoding with ridge regression on the CNeuroMod Friends dataset, one of the largest deep fMRI resource currently available. With multi-threading, our results show that the Intel Math Kernel Library (MKL) significantly outperforms the OpenBLAS library, being 1.9 times faster using 32 threads on a single machine. We then evaluated the Dask multi-CPU implementation of ridge regression readily available in scikit-learn (MultiOutput), and we proposed a new "batch" version of Dask parallelization, motivated by a time complexity analysis. In line with our theoretical analysis, MultiOutput parallelization was found to be impractical, i.e., slower than multi-threading on a single machine. In contrast, the Batch-MultiOutput regression scaled well across compute nodes and threads, providing speed-ups of up to 33 times with 8 compute nodes and 32 threads compared to a single-threaded scikit-learn execution. Batch parallelization using Dask thus emerges as a scalable approach for brain encoding with ridge regression on high-performance computing systems using scikit-learn and large fMRI datasets.

We propose a deep learning algorithm for the inpainting of Bragg Coherent Diffraction Imaging (BCDI) patterns affected by detector gaps. These regions of missing intensity can compromise the accuracy of reconstruction algorithms, inducing artifacts in the final result. It is thus desirable to restore the intensity in these regions in order to ensure more reliable reconstructions. The key aspect of our method lies in the choice of training the neural network with cropped sections of both experimental diffraction data and simulated data and subsequently patching the predictions generated by the model along the gap, thus completing the full diffraction peak. This provides us with more experimental training data and allows for a faster model training due to the limited size, while the neural network can be applied to arbitrarily larger BCDI datasets. Moreover, our method not only broadens the scope of application but also ensures the preservation of data integrity and reliability in the face of challenging experimental conditions., Comment: Main article + Supplemental Material

Neurons in the brain communicate information via punctual events called spikes. The timing of spikes is thought to carry rich information, but it is not clear how to leverage this in digital systems. We demonstrate that event-based encoding is efficient for audio compression. To build this event-based representation we use a deep binary auto-encoder, and under high sparsity pressure, the model enters a regime where the binary event matrix is stored more efficiently with sparse matrix storage algorithms. We test this on the large MAESTRO dataset of piano recordings against vector quantized auto-encoders. Not only does our "Spiking Music compression" algorithm achieve a competitive compression/reconstruction trade-off, but selectivity and synchrony between encoded events and piano key strikes emerge without supervision in the sparse regime.

We consider unregularized robust M-estimators for linear models under Gaussian design and heavy-tailed noise, in the proportional asymptotics regime where the sample size n and the number of features p are both increasing such that $p/n \to \gamma\in (0,1)$. An estimator of the out-of-sample error of a robust M-estimator is analysed and proved to be consistent for a large family of loss functions that includes the Huber loss. As an application of this result, we propose an adaptive tuning procedure of the scale parameter $\lambda>0$ of a given loss function $\rho$: choosing$\hat \lambda$ in a given interval $I$ that minimizes the out-of-sample error estimate of the M-estimator constructed with loss $\rho_\lambda(\cdot) = \lambda^2 \rho(\cdot/\lambda)$ leads to the optimal out-of-sample error over $I$. The proof relies on a smoothing argument: the unregularized M-estimation objective function is perturbed, or smoothed, with a Ridge penalty that vanishes as $n\to+\infty$, and show that the unregularized M-estimator of interest inherits properties of its smoothed version., Comment: 33 pages, 10 figures

Major progress has been made in the previous decade to characterize the asymptotic behavior of regularized M-estimators in high-dimensional regression problems in the proportional asymptotic regime where the sample size $n$ and the number of features $p$ are increasing simultaneously such that $n/p\to \delta \in(0,\infty)$, using powerful tools such as Approximate Message Passing or the Convex Gaussian Min-Max Theorem (CGMT). The asymptotic error and behavior of the regularized M-estimator is then typically described by a system of nonlinear equations with a few scalar unknowns, and the solution to this system precisely characterizes the asymptotic error. Application of the CGMT and related machinery requires the existence and uniqueness of a solution to this low-dimensional system of equations or to a related scalar convex minimization problem. This paper resolves the question of existence and uniqueness of solution to this low-dimensional system for the case of linear models with independent additive noise, when both the data-fitting loss function and regularizer are separable and convex. Such existence result was previously known under strong convexity or for specific estimators such as the Lasso. The main idea behind this existence result is inspired by an argument developed by Montanari et al. [2023] and Celentano et al. [2023] in different contexts: By constructing an ad-hoc convex minimization problem in an infinite dimensional Hilbert space, the existence of the Lagrange multiplier for this optimization problem makes it possible to construct explicit solutions to the low-dimensional system of interest. The conditions under which we derive this existence result exactly correspond to the side of the phase transition where perfect recovery $\hat{x} = x_0$ fails, so that these conditions are optimal., Comment: 55 pages, 5 figures

Generalized cross-validation (GCV) is a widely-used method for estimating the squared out-of-sample prediction risk that employs a scalar degrees of freedom adjustment (in a multiplicative sense) to the squared training error. In this paper, we examine the consistency of GCV for estimating the prediction risk of arbitrary ensembles of penalized least-squares estimators. We show that GCV is inconsistent for any finite ensemble of size greater than one. Towards repairing this shortcoming, we identify a correction that involves an additional scalar correction (in an additive sense) based on degrees of freedom adjusted training errors from each ensemble component. The proposed estimator (termed CGCV) maintains the computational advantages of GCV and requires neither sample splitting, model refitting, or out-of-bag risk estimation. The estimator stems from a finer inspection of the ensemble risk decomposition and two intermediate risk estimators for the components in this decomposition. We provide a non-asymptotic analysis of the CGCV and the two intermediate risk estimators for ensembles of convex penalized estimators under Gaussian features and a linear response model. Furthermore, in the special case of ridge regression, we extend the analysis to general feature and response distributions using random matrix theory, which establishes model-free uniform consistency of CGCV., Comment: 91 pages, 34 figures; this version adds general proof outlines (in Sections 4.3 and 5.3), add more experiments with non-Gaussian data (in Sections D and E), relaxes an assumption (in Section A.7), clarifies explanations at several places, and corrects minor typos at several places

Ultrafast X-ray diffraction by the LCLS free-electron laser has been used to probe Charge Density Wave (CDW) systems under applied external currents. At sufficiently low currents, CDW wavefronts bend in the direction transverse to the 2k$_F$ wave vector. We show that this shear effect has the ability to focus or defocus hard X-ray beams, depending of the current direction, making it an electronic lens of a new kind, tunable at will from the Fraunhofer to the Fresnel regime. The effect is interpreted using the fractional Fourier transform showing how the macroscopic curvature of a nanometric modulation can be beneficially used to modify the propagation of X-ray beams., Comment: 3 pages, 3 figures

In this article, we review different studies based on advanced x-ray diffraction techniques - especially coherent x-ray diffraction - that allowed us to reveal the behaviour of such symmetry-breaking systems as Charge Density Wave (CDW) and Spin density Wave (SDW), through their local phase. After a brief introduction on the added value of using coherent x-rays, we show how the method can be applied to CDW and SDW systems, in both static and dynamical regimes. The approach allowed us to probe the particular sliding state of CDWs systems by observing them through their phase fluctuations, to which coherent x-rays are particularly sensitive. Several compounds stabilizing a CDW phase able to slide are presented, each with a different but clearly pronounced signature of the sliding state. Two main features emerge from this series of experiments which have been little treated until now, the influence of CDW pinning by the sample surfaces and the propagation of periodic phase defects such as charge solitons across the entire sample. Phase models describing the spatial and temporal properties of sliding CDWs are presented in the last part of this review., Comment: 41 pages, 20 figures

Ensuring the long-term reproducibility of data analyses requires results stability tests to verify that analysis results remain within acceptable variation bounds despite inevitable software updates and hardware evolutions. This paper introduces a numerical variability approach for results stability tests, which determines acceptable variation bounds using random rounding of floating-point calculations. By applying the resulting stability test to \fmriprep, a widely-used neuroimaging tool, we show that the test is sensitive enough to detect subtle updates in image processing methods while remaining specific enough to accept numerical variations within a reference version of the application. This result contributes to enhancing the reliability and reproducibility of data analyses by providing a robust and flexible method for stability testing.

The electronic orders appearing in condensed matter systems are originating from the precise arrangement of atoms constituting the crystal as well as their nature. This teneous relationship can lead to highly different phases in condensed matter, and drive electronic phase transitions. Here, we show that a very slight deformation of the crystal structure of TbTe$_3$ can have a dramatic influence on the electronic order that is stabilized. In particular, we show that the Charge Density Wave (CDW) developping along the $\vec{c}$ axis in the pristine state, switches to an orientation along $\vec{a}$ when the naturally orthorhombic system is turned into a tetragonal system. This is achieved by performing true biaxial mechanical deformation of a TbTe$_3$ sample from 250K to 375K, and by measuring both structural and electronic parameters with x-ray diffraction and transport measurements. We show that this switching transition is driven by the tetragonality parameter $a/c$, and that the transition occurs for $a=c$, with a coexistence region for $0.9985< a/c < 1.002$. The CDW transition temperature $T_c$ is found to have a linear dependence with $a/c$, with no saturation in the deformed states investigated here, while the gap saturates out of the coexistence region. The linear dependence of $T_c$ is accounted for within a tight-binding model. Our results question the relationship between the gap and $T_c$ in RTe$_3$ systems. More generally, our method of applying true biaxial deformation at cryogenic temperatures can be applied to many systems displaying electronic phase transitions, and opens a new route towards the study of coexisting or competing electronic orders in condensed matter.

Communication by rare, binary spikes is a key factor for the energy efficiency of biological brains. However, it is harder to train biologically-inspired spiking neural networks (SNNs) than artificial neural networks (ANNs). This is puzzling given that theoretical results provide exact mapping algorithms from ANNs to SNNs with time-to-first-spike (TTFS) coding. In this paper we analyze in theory and simulation the learning dynamics of TTFS-networks and identify a specific instance of the vanishing-or-exploding gradient problem. While two choices of SNN mappings solve this problem at initialization, only the one with a constant slope of the neuron membrane potential at threshold guarantees the equivalence of the training trajectory between SNNs and ANNs with rectified linear units. We demonstrate that training deep SNN models achieves the exact same performance as that of ANNs, surpassing previous SNNs on image classification datasets such as MNIST/Fashion-MNIST, CIFAR10/CIFAR100 and PLACES365. Our SNN accomplishes high-performance classification with less than 0.3 spikes per neuron, lending itself for an energy-efficient implementation. We show that fine-tuning SNNs with our robust gradient descent algorithm enables their optimization for hardware implementations with low latency and resilience to noise and quantization.

Simultaneous behavioral and electrophysiological recordings call for new methods to reveal the interactions between neural activity and behavior. A milestone would be an interpretable model of the co-variability of spiking activity and behavior across trials. Here, we model a mouse cortical sensory-motor pathway in a tactile detection task reported by licking with a large recurrent spiking neural network (RSNN), fitted to the recordings via gradient-based optimization. We focus specifically on the difficulty to match the trial-to-trial variability in the data. Our solution relies on optimal transport to define a distance between the distributions of generated and recorded trials. The technique is applied to artificial data and neural recordings covering six cortical areas. We find that the resulting RSNN can generate realistic cortical activity and predict jaw movements across the main modes of trial-to-trial variability. Our analysis also identifies an unexpected mode of variability in the data corresponding to task-irrelevant movements of the mouse., Comment: 12 pages of main text, 4 figures in main, 5 pages of appendix, 5 figures in appendix

"You never forget how to ride a bike", -- but how is that possible? The brain is able to learn complex skills, stop the practice for years, learn other skills in between, and still retrieve the original knowledge when necessary. The mechanisms of this capability, referred to as lifelong learning (or continual learning, CL), are unknown. We suggest a bio-plausible meta-plasticity rule building on classical work in CL which we summarize in two principles: (i) neurons are context selective, and (ii) a local availability variable partially freezes the plasticity if the neuron was relevant for previous tasks. In a new neuro-centric formalization of these principles, we suggest that neuron selectivity and neuron-wide consolidation is a simple and viable meta-plasticity hypothesis to enable CL in the brain. In simulation, this simple model balances forgetting and consolidation leading to better transfer learning than contemporary CL algorithms on image recognition and natural language processing CL benchmarks.

This paper investigates the asymptotic distribution of the maximum-likelihood estimate (MLE) in multinomial logistic models in the high-dimensional regime where dimension and sample size are of the same order. While classical large-sample theory provides asymptotic normality of the MLE under certain conditions, such classical results are expected to fail in high-dimensions as documented for the binary logistic case in the seminal work of Sur and Cand\`es [2019]. We address this issue in classification problems with 3 or more classes, by developing asymptotic normality and asymptotic chi-square results for the multinomial logistic MLE (also known as cross-entropy minimizer) on null covariates. Our theory leads to a new methodology to test the significance of a given feature. Extensive simulation studies on synthetic data corroborate these asymptotic results and confirm the validity of proposed p-values for testing the significance of a given feature.

There is a recent surge in interest for imitation learning, with large human video-game and robotic manipulation datasets being used to train agents on very complex tasks. While deep neuroevolution has recently been shown to match the performance of gradient-based techniques on various reinforcement learning problems, the application of deep neuroevolution techniques to imitation learning remains relatively unexplored. In this work, we propose to explore whether deep neuroevolution can be used for behaviour imitation on popular simulation environments. We introduce a simple co-evolutionary adversarial generation framework, and evaluate its capabilities by evolving standard deep recurrent networks to imitate state-of-the-art pre-trained agents on 8 OpenAI Gym state-based control tasks. Across all tasks, we find the final elite actor agents capable of achieving scores as high as those obtained by the pre-trained agents, all the while closely following their score trajectories. Our results suggest that neuroevolution could be a valuable addition to deep learning techniques to produce accurate emulation of behavioural agents. We believe that the generality and simplicity of our approach opens avenues for imitating increasingly complex behaviours in increasingly complex settings, e.g. human behaviour in real-world settings. We provide our source code, model checkpoints and results at github.com/MaximilienLC/gane.

Modern artificial intelligence works typically train the parameters of fixed-sized deep neural networks using gradient-based optimization techniques. Simple evolutionary algorithms have recently been shown to also be capable of optimizing deep neural network parameters, at times matching the performance of gradient-based techniques, e.g. in reinforcement learning settings. In addition to optimizing network parameters, many evolutionary computation techniques are also capable of progressively constructing network architectures. However, constructing network architectures from elementary evolution rules has not yet been shown to scale to modern reinforcement learning benchmarks. In this paper we therefore propose a new approach in which the architectures of recurrent neural networks dynamically evolve according to a small set of mutation rules. We implement a massively parallel evolutionary algorithm and run experiments on all 19 OpenAI Gym state-based reinforcement learning control tasks. We find that in most cases, dynamic agents match or exceed the performance of gradient-based agents while utilizing orders of magnitude fewer parameters. We believe our work to open avenues for real-life applications where network compactness and autonomous design are of critical importance. We provide our source code, final model checkpoints and full results at github.com/MaximilienLC/nra.

Deep spiking neural networks (SNNs) offer the promise of low-power artificial intelligence. However, training deep SNNs from scratch or converting deep artificial neural networks to SNNs without loss of performance has been a challenge. Here we propose an exact mapping from a network with Rectified Linear Units (ReLUs) to an SNN that fires exactly one spike per neuron. For our constructive proof, we assume that an arbitrary multi-layer ReLU network with or without convolutional layers, batch normalization and max pooling layers was trained to high performance on some training set. Furthermore, we assume that we have access to a representative example of input data used during training and to the exact parameters (weights and biases) of the trained ReLU network. The mapping from deep ReLU networks to SNNs causes zero percent drop in accuracy on CIFAR10, CIFAR100 and the ImageNet-like data sets Places365 and PASS. More generally our work shows that an arbitrary deep ReLU network can be replaced by an energy-efficient single-spike neural network without any loss of performance.

We demonstrate, based on low-temperature scanning tunneling microscopy (STM) and spectroscopy, a pronounced negative differential resistance (NDR) in spin-crossover (SCO) molecular devices, where a Fe$^{\text{II}}$ SCO molecule is deposited on surfaces. The STM measurements reveal that the NDR is robust with respect to substrate materials, temperature, and the number of SCO layers. This indicates that the NDR is intrinsically related to the electronic structure of the SCO molecule. Experimental results are supported by density functional theory (DFT) with non-equilibrium Green's functions (NEGF) calculations and a generic theoretical model. While the DFT+NEGF calculations reproduce NDR for a special atomically-sharp STM tip, the effect is attributed to the energy-dependent tip density of states rather than the molecule itself. We, therefore, propose a Coulomb blockade model involving three molecular orbitals with very different spatial localization as suggested by the molecular electronic structure., Comment: 4 figures

This paper studies the multi-task high-dimensional linear regression models where the noise among different tasks is correlated, in the moderately high dimensional regime where sample size $n$ and dimension $p$ are of the same order. Our goal is to estimate the covariance matrix of the noise random vectors, or equivalently the correlation of the noise variables on any pair of two tasks. Treating the regression coefficients as a nuisance parameter, we leverage the multi-task elastic-net and multi-task lasso estimators to estimate the nuisance. By precisely understanding the bias of the squared residual matrix and by correcting this bias, we develop a novel estimator of the noise covariance that converges in Frobenius norm at the rate $n^{-1/2}$ when the covariates are Gaussian. This novel estimator is efficiently computable. Under suitable conditions, the proposed estimator of the noise covariance attains the same rate of convergence as the "oracle" estimator that knows in advance the regression coefficients of the multi-task model. The Frobenius error bounds obtained in this paper also illustrate the advantage of this new estimator compared to a method-of-moments estimator that does not attempt to estimate the nuisance. As a byproduct of our techniques, we obtain an estimate of the generalization error of the multi-task elastic-net and multi-task lasso estimators. Extensive simulation studies are carried out to illustrate the numerical performance of the proposed method.

Can we use spiking neural networks (SNN) as generative models of multi-neuronal recordings, while taking into account that most neurons are unobserved? Modeling the unobserved neurons with large pools of hidden spiking neurons leads to severely underconstrained problems that are hard to tackle with maximum likelihood estimation. In this work, we use coarse-graining and mean-field approximations to derive a bottom-up, neuronally-grounded latent variable model (neuLVM), where the activity of the unobserved neurons is reduced to a low-dimensional mesoscopic description. In contrast to previous latent variable models, neuLVM can be explicitly mapped to a recurrent, multi-population SNN, giving it a transparent biological interpretation. We show, on synthetic spike trains, that a few observed neurons are sufficient for neuLVM to perform efficient model inversion of large SNNs, in the sense that it can recover connectivity parameters, infer single-trial latent population activity, reproduce ongoing metastable dynamics, and generalize when subjected to perturbations mimicking photo-stimulation., Comment: 23 pages, 7 figures

Large scale structures in the Universe, ranging from globular clusters to entire galaxies, are the manifestation of relaxation to out-of-equilibrium states that are not described by standard statistical mechanics at equilibrium. Instead, they are formed through a process of a very different nature, i.e. violent relaxation. However, astrophysical time-scales are so large that it is not possible to directly observe these relaxation dynamics and therefore verify the details of the violent relaxation process. We develop a table-top experiment and model that allows us to directly observe effects such as mixing of phase space, and violent relaxation, leading to the formation of a table-top analogue of a galaxy. The experiment allows us to control a range of parameters, including the nonlocal (gravitational) interaction strength and quantum effects, thus providing an effective test-bed for gravitational models that cannot otherwise be directly studied in experimental settings., Comment: 7 pages, 5 figures; Supplementary Information 9 pages and 7 figures

We consider observations $(X,y)$ from single index models with unknown link function, Gaussian covariates and a regularized M-estimator $\hat\beta$ constructed from convex loss function and regularizer. In the regime where sample size $n$ and dimension $p$ are both increasing such that $p/n$ has a finite limit, the behavior of the empirical distribution of $\hat\beta$ and the predicted values $X\hat\beta$ has been previously characterized in a number of models: The empirical distributions are known to converge to proximal operators of the loss and penalty in a related Gaussian sequence model, which captures the interplay between ratio $p/n$, loss, regularization and the data generating process. This connection between$(\hat\beta,X\hat\beta)$ and the corresponding proximal operators require solving fixed-point equations that typically involve unobservable quantities such as the prior distribution on the index or the link function. This paper develops a different theory to describe the empirical distribution of $\hat\beta$ and $X\hat\beta$: Approximations of $(\hat\beta,X\hat\beta)$ in terms of proximal operators are provided that only involve observable adjustments. These proposed observable adjustments are data-driven, e.g., do not require prior knowledge of the index or the link function. These new adjustments yield confidence intervals for individual components of the index, as well as estimators of the correlation of $\hat\beta$ with the index. The interplay between loss, regularization and the model is thus captured in a data-driven manner, without solving the fixed-point equations studied in previous works. The results apply to both strongly convex regularizers and unregularized M-estimation. Simulations are provided for the square and logistic loss in single index models including logistic regression and 1-bit compressed sensing with 20\% corrupted bits.

The paper proposes chi-square and normal inference methodologies for the unknown coefficient matrix $B^*$ of size $p\times T$ in a Multi-Task (MT) linear model with $p$ covariates, $T$ tasks and $n$ observations under a row-sparse assumption on $B^*$. The row-sparsity $s$, dimension $p$ and number of tasks $T$ are allowed to grow with $n$. In the high-dimensional regime $p\ggg n$, in order to leverage row-sparsity, the MT Lasso is considered. We build upon the MT Lasso with a de-biasing scheme to correct for the bias induced by the penalty. This scheme requires the introduction of a new data-driven object, coined the interaction matrix, that captures effective correlations between noise vector and residuals on different tasks. This matrix is psd, of size $T\times T$ and can be computed efficiently. The interaction matrix lets us derive asymptotic normal and $\chi^2_T$ results under Gaussian design and $\frac{sT+s\log(p/s)}{n}\to0$ which corresponds to consistency in Frobenius norm. These asymptotic distribution results yield valid confidence intervals for single entries of $B^*$ and valid confidence ellipsoids for single rows of $B^*$, for both known and unknown design covariance $\Sigma$. While previous proposals in grouped-variables regression require row-sparsity $s\lesssim\sqrt n$ up to constants depending on $T$ and logarithmic factors in $n,p$, the de-biasing scheme using the interaction matrix provides confidence intervals and $\chi^2_T$ confidence ellipsoids under the conditions ${\min(T^2,\log^8p)}/{n}\to 0$ and $$ \frac{sT+s\log(p/s)+\|\Sigma^{-1}e_j\|_0\log p}{n}\to0, \quad \frac{\min(s,\|\Sigma^{-1}e_j\|_0)}{\sqrt n} \sqrt{[T+\log(p/s)]\log p}\to 0, $$ allowing row-sparsity $s\ggg\sqrt n$ when $\|\Sigma^{-1}e_j\|_0 \sqrt T\lll \sqrt{n}$ up to logarithmic factors.

This paper studies M-estimators with gradient-Lipschitz loss function regularized with convex penalty in linear models with Gaussian design matrix and arbitrary noise distribution. A practical example is the robust M-estimator constructed with the Huber loss and the Elastic-Net penalty and the noise distribution has heavy-tails. Our main contributions are three-fold. (i) We provide general formulae for the derivatives of regularized M-estimators $\hat\beta(y,X)$ where differentiation is taken with respect to both $y$ and $X$; this reveals a simple differentiability structure shared by all convex regularized M-estimators. (ii) Using these derivatives, we characterize the distribution of the residual $r_i = y_i-x_i^\top\hat\beta$ in the intermediate high-dimensional regime where dimension and sample size are of the same order. (iii) Motivated by the distribution of the residuals, we propose a novel adaptive criterion to select tuning parameters of regularized M-estimators. The criterion approximates the out-of-sample error up to an additive constant independent of the estimator, so that minimizing the criterion provides a proxy for minimizing the out-of-sample error. The proposed adaptive criterion does not require the knowledge of the noise distribution or of the covariance of the design. Simulated data confirms the theoretical findings, regarding both the distribution of the residuals and the success of the criterion as a proxy of the out-of-sample error. Finally our results reveal new relationships between the derivatives of $\hat\beta(y,X)$ and the effective degrees of freedom of the M-estimator, which are of independent interest.

This paper develops asymptotic normality results for individual coordinates of robust M-estimators with convex penalty in high-dimensions, where the dimension $p$ is at most of the same order as the sample size $n$, i.e, $p/n\le\gamma$ for some fixed constant $\gamma>0$. The asymptotic normality requires a bias correction and holds for most coordinates of the M-estimator for a large class of loss functions including the Huber loss and its smoothed versions regularized with a strongly convex penalty. The asymptotic variance that characterizes the width of the resulting confidence intervals is estimated with data-driven quantities. This estimate of the variance adapts automatically to low ($p/n\to0)$ or high ($p/n \le \gamma$) dimensions and does not involve the proximal operators seen in previous works on asymptotic normality of M-estimators. For the Huber loss, the estimated variance has a simple expression involving an effective degrees-of-freedom as well as an effective sample size. The case of the Huber loss with Elastic-Net penalty is studied in details and a simulation study confirms the theoretical findings. The asymptotic normality results follow from Stein formulae for high-dimensional random vectors on the sphere developed in the paper which are of independent interest.