arXiv subjects

Ionel Popescu

Publications and source records attributed to Ionel Popescu.

At least 19 recordsLinked to original sources

arXiv2026

One-Cut Risk Profiles under Quadratic Loss: Discrete Convexity, Continuous Limits, and Higher Dimensions

In this note we study a two-regime representation of a loss random variable under quadratic error. For a finite law we compute exactly the change of the optimal risk when one atom crosses the cut. This turns the problem into a convexity question in cumulative-mass coordinates. On an equally spaced support, log-concavity gives this convexity, while weak symmetry locates the optimal cut, with an additional correction when the mean lies between two atoms. We also discuss the continuous analogue and extend the main identities to finitely supported random vectors, where a global optimal partition may be chosen as a halfspace.

math.PR↗

arXiv2026

Non-Gaussianity of the Stagnation Law in Particle Swarm Optimization

We study one-dimensional particle swarm optimization during stagnation, with two fixed distinct attractors and equal independent uniform acceleration ranges. The position then satisfies a second-order random affine recurrence. For inertia $w$ and acceleration range $c$, we prove that throughout the open mean-square stability region \[ -1 0,\qquad 12(1-w^2)-c(7-5w)>0, \] no invariant position marginal, and hence no limiting position marginal, can be Gaussian. This solves the open Problem 18 in \cite{ParticleSwarmProblems}. The proof compares the stationary moment equations with the Gaussian moment identities through order eight. A Hermite-polynomial formulation gives explicit fourth- and sixth-order compatibility conditions whose common solutions lie on a degree-$107$ polynomial branch. Exact eighth-order equations exclude every point on that branch. The final certificate is verified using arithmetic modulo $23$ and independently modulo $1{,}000{,}003$. A separate raw-moment implementation produces exact polynomials $Q_4,Q_6,Q_8$ in $(w,c)$ and verifies the same obstruction over the rational numbers. The fourth- and sixth-order curves have a genuine admissible intersection, but the eighth-order condition removes it, showing why low-order Gaussian diagnostics are insufficient. All code, exact polynomials, logs, and plot-validation data are supplied as online resources.

math.PR↗

arXiv2026

Rank-One Fluctuations in Averaging-Learning Dynamics

We study averaging-learning dynamics without an exogenous ground truth: the reference signal is generated endogenously by the population. The dynamics combine a time-varying averaging matrix, a learning-source matrix, and a learning matrix, typically diagonal. Dobrushin-type contraction controls the decay of oscillations and yields asymptotic agreement. Under a summability condition, the backward products converge exponentially to rank-one limits. With summable perturbations, the process converges to a random consensus state. For i.i.d. perturbations, we prove a central limit theorem for the centered process: the limiting Gaussian law is supported on the agreement direction. Thus, despite the multi-agent dynamics, the long-time fluctuations are asymptotically one-dimensional. We also record a pairwise Dobrushin formulation that clarifies the dynamic agreement-class geometry underlying the one-class regime.

math.PR↗

arXiv2026

A property of log-concave and weakly-symmetric distributions for two step approximations of random variables

In this paper we introduce a generalization of classical risk measures in which the risk is represented by a step function taking two values, corresponding to two endogenously determined market regimes. This extends the traditional framework where risk measures map random variables to single real numbers. For the quadratic loss function, we study the optimization problem of determining the optimal regime threshold and corresponding values. In the case of log-concave distributions we give conditions for the uniqueness of the regime changing. We treat the case of one dimension and also of multi-dimensions for elliptic distributions. We demonstrate the necessity of convexity through counterexamples.

math.PR↗

arXiv2026

Anchoring and Mixed-Norm Contractions in Averaging-Learning Dynamics

A single informed agent can draw an arbitrarily large network to the ground truth. This is the sharpest consequence of the "Averaging plus Learning" framework studied here, where agents update opinions by socially averaging neighbours while some receive private feedback at heterogeneous rates. The key is a graph-theoretic property we call condensely anchored, which implies convergence to the correct consensus on fixed networks. In the original framework of Popescu and Vaidya (2023), every agent was required to learn. Removing that requirement changes the problem fundamentally: the underlying graph must now carry the signal from a handful of anchors to everyone else. When learning rates decay to zero, a persistence condition on the rates alone suffices, with no uniform connectivity or aperiodicity assumed. The hardest case is intermittent connectivity, where no single time step contracts in any standard norm. A mixed-operator-norm framework is developed that extracts two-step contraction from the interplay between aggregate learning mass and entrywise diffusion of influence, a mechanism new to consensus literature. Finally, we demonstrate the framework's robustness: vanishing noise preserves convergence to the ground truth, whereas persistent noise drives the system to a limiting law.

math.DS↗

arXiv2025

Tridiagonal random matrices, an analytic approach

In this paper, we study the limiting distribution of the eigenvalues for random tridiagonal matrix models. In the paper \cite{P09}, the limiting distribution is well described by its moments. Here, an analytical approach allows us, as in the case of Wigner matrices, to relax the assumptions on the random variables. With this method, we proved the convergence of the spectral distribution under an assumption on the second moment. We discuss also about an algebraic approach for the tridiagonal models, which are more complicated than the classic freeness.

math.PR↗

arXiv2025

Quantitative boundary H\"{o}lder estimates for the inhomogeneous Poisson problem through a probabilistic approach

In this paper we derive quantitative boundary H\"older estimates, with explicit constants, for the inhomogeneous Poisson problem in a bounded open set $D\subset \mathbb{R}^d$. Our approach has two main steps: firstly, we consider an arbitrary $D$ as above and prove that the boundary $\alpha$-H\"older regularity of the solution the Poisson equation is controlled, with explicit constants, by the H\"older seminorm of the boundary data, the $L^ \gamma$-norm of the forcing term with $\gamma>d/2$, and the $\alpha/2$-moment of the exit time from $D$ of the Brownian motion. Secondly, we derive explicit estimates for the $\alpha/2$-moment of the exit time in terms of the distance to the boundary, the regularity of the domain $D$, and $\alpha$. Using this approach, we derive explicit estimates for the same problem in domains satisfying exterior ball conditions, respectively exterior cone/wedge conditions, in terms of simple geometric features. As a consequence we also obtain explicit constants for pointwise estimates for the Green function and for the gradient of the solution. The obtained estimates can be employed to bypass the curse of high dimensions when aiming to approximate the solution of the Poisson problem using neural networks, obtaining polynomial scaling with dimension, which in some cases can be shown to be optimal.

math.PR↗

arXiv2025

Error analysis for the deep Kolmogorov method

The deep Kolmogorov method is a simple and popular deep learning based method for approximating solutions of partial differential equations (PDEs) of the Kolmogorov type. In this work we provide an error analysis for the deep Kolmogorov method for heat PDEs. Specifically, we reveal convergence with convergence rates for the overall mean square distance between the exact solution of the heat PDE and the realization function of the approximating deep neural network (DNN) associated with a stochastic optimization algorithm in terms of the size of the architecture (the depth/number of hidden layers and the width of the hidden layers) of the approximating DNN, in terms of the number of random sample points used in the loss function (the number of input-output data pairs used in the loss function), and in terms of the size of the optimization error made by the employed stochastic optimization method.

math.NA↗

arXiv2024

Generic local identifiability for ODE inverse problems from discrete observations

We study local identifiability of parameters in ordinary differential equation models from finitely many observations. The central object is the parameter-to-observation map obtained by sampling the solution at prescribed times. We first prove a quantitative injectivity estimate for general $C^2$ observation maps: a lower bound on the smallest singular value of the parameter Jacobian, together with an upper bound on the second derivative, gives an explicit neighborhood on which the inverse problem has a unique and stable local solution. We then treat analytic ODE models. For analytic vector fields the observation map is analytic in observation times, initial states, and parameters; consequently, the loss of full parameter rank is contained in the zero set of a real analytic function. Under a single non-degeneracy condition this gives generic local identifiability, including for randomly chosen observation times with a density. Finally, for homogeneous linear systems $\dot X=AX$, we separate the recovery of $e^{hA}$ from the recovery of $A$: cyclic initial states identify the discrete propagator from one trajectory, while the remaining ambiguity is precisely the ambiguity of the real matrix logarithm.

math.CA↗

arXiv2023

Approximation and interpolation of deep neural networks

In this paper, we prove that in the overparametrized regime, deep neural network provide universal approximations and can interpolate any data set, as long as the activation function is locally in $L^1(\RR)$ and not an affine function. Additionally, if the activation function is smooth and such an interpolation networks exists, then the set of parameters which interpolate forms a manifold. Furthermore, we give a characterization of the Hessian of the loss function evaluated at the interpolation points. In the last section, we provide a practical probabilistic method of finding such a point under general conditions on the activation function.

cs.LG↗

arXiv2022

From Monte Carlo to neural networks approximations of boundary value problems

In this paper we study probabilistic and neural network approximations for solutions to Poisson equation subject to Holder data in general bounded domains of $\mathbb{R}^d$. We aim at two fundamental goals. The first, and the most important, we show that the solution to Poisson equation can be numerically approximated in the sup-norm by Monte Carlo methods, and that this can be done highly efficiently if we use a modified version of the walk on spheres algorithm as an acceleration method. This provides estimates which are efficient with respect to the prescribed approximation error and with polynomial complexity in the dimension and the reciprocal of the error. A crucial feature is that the overall number of samples does not not depend on the point at which the approximation is performed. As a second goal, we show that the obtained Monte Carlo solver renders in a constructive way ReLU deep neural network (DNN) solutions to Poisson problem, whose sizes depend at most polynomialy in the dimension $d$ and in the desired error. In fact we show that the random DNN provides with high probability a small approximation error and low polynomial complexity in the dimension.

math.PR↗

arXiv2022

LoCoV: low dimension covariance voting algorithm for portfolio optimization

Minimum-variance portfolio optimizations rely on accurate covariance estimator to obtain optimal portfolios. However, it usually suffers from large error from sample covariance matrix when the sample size $n$ is not significantly larger than the number of assets $p$. We analyze the random matrix aspects of portfolio optimization and identify the order of errors in sample optimal portfolio weight and show portfolio risk are underestimated when using samples. We also provide LoCoV (low dimension covariance voting) algorithm to reduce error inherited from random samples. From various experiments, LoCoV is shown to outperform the classical method by a large margin.

q-fin.PM↗

arXiv2022

Inverse problem for parameters identification in a modified SIRD epidemic model using ensemble neural networks

In this paper, we propose a parameter identification methodology of the SIRD model, an extension of the classical SIR model, that considers the deceased as a separate category. In addition, our model includes one parameter which is the ratio between the real total number of infected and the number of infected that were documented in the official statistics. Due to many factors, like governmental decisions, several variants circulating, opening and closing of schools, the typical assumption that the parameters of the model stay constant for long periods of time is not realistic. Thus our objective is to create a method which works for short periods of time. In this scope, we approach the estimation relying on the previous 7 days of data and then use the identified parameters to make predictions. To perform the estimation of the parameters we propose the average of an ensemble of neural networks. Each neural network is constructed based on a database built by solving the SIRD for 7 days, with random parameters. In this way, the networks learn the parameters from the solution of the SIRD model. Lastly we use the ensemble to get estimates of the parameters from the real data of Covid19 in Romania and then we illustrate the predictions for different periods of time, from 10 up to 45 days, for the number of deaths. The main goal was to apply this approach on the analysis of COVID-19 evolution in Romania, but this was also exemplified on other countries like Hungary, Czech Republic and Poland with similar results. The results are backed by a theorem which guarantees that we can recover the parameters of the model from the reported data. We believe this methodology can be used as a general tool for dealing with short term predictions of infectious diseases or in other compartmental models.

cs.LG↗

arXiv2022

Recover the spectrum of covariance matrix: a non-asymptotic iterative method

It is well known the sample covariance has a consistent bias in the spectrum, for example spectrum of Wishart matrix follows the Marchenko-Pastur law. We in this work introduce an iterative algorithm 'Concent' that actively eliminate this bias and recover the true spectrum for small and moderate dimensions.

stat.ML↗

arXiv2021

Invariance principle of random projection for the norm

Johnson-Lindenstrauss guarantees certain topological structure is preserved under random projections when project high dimensional deterministic vectors to low dimensional vectors. In this work, we try to understand how random matrix affect norms of random vectors. In particular we prove the distribution of the norm of random vector $X \in \mathbb{R}^n$, whose entries are i.i.d. random variables, is preserved by random projection $S:\mathbb{R}^n \to \mathbb{R}^m$. More precisely, \[ \frac{X^TS^TSX - mn}{\sqrt{\sigma^2 m^2n+2mn^2}} \xrightarrow[\quad m/n\to 0 \quad ]{ m,n\to \infty } \mathcal{N}(0,1) \] We also prove a concentration of the random norm transformed by either random projection or random embedding. Overall, our results showed random matrix has low distortion for the norm of random vectors with i.i.d. entries.

math.PR↗

arXiv2021

A product-CLT and its application in invariance principle of random projection

Johnson-Lindenstrauss lemma states random projections can be used as a topology preserving embedding technique for fixed vectors. In this paper, we try to understand how random projections affect probabilistic properties of random vectors. In particular we prove the distribution of inner product of two independent random vectors $X, Z \in {R}^n$ is preserved by random projection $S:{R}^n \to {R}^m$. More precisely, \[ \sup_t \left| \text{P}(\frac{1}{C_{m,n}} X^TS^TSZ <t) - \text{P}(\frac{1}{\sqrt{n}} X^TZ<t) \right| \le O\left(\frac{1}{\sqrt{n}}+ \frac{1}{\sqrt{m}} \right) \] This is achieved by proving a general central limit theorem (product-CLT) for $\sum_{k=1}^{n} X_k Y_k$, where $\{X_k\}$ is a martingale difference sequence, and $\{Y_k\}$ has dependency within the sequence. We also obtain the rate of convergence in the spirit of Berry-Esseen theorem.

math.PR↗

arXiv2020

A regime switching on Covid19 analysis and prediction in Romania

In this paper we propose a three stages analysis of the evolution of Covid19 in Romania. There are two main issues when it comes to pandemic prediction. The first one is the fact that the numbers reported of infected and recovered are unreliable, however the number of deaths is more accurate. The second issue is that there were many factors which affected the evolution of the pandemic. In this paper we propose an analysis in three stages. The first stage is based on the classical SIR model which we do using a neural network. This provides a first set of daily parameters. In the second stage we propose a refinement of the SIR model in which we separate the deceased into a distinct category. By using the first estimate and a grid search, we give a daily estimation of the parameters. The third stage is used to define a notion of turning points (local extremes) for the parameters. We call a regime the time between these points. We outline a general way based on time varying parameters of SIRD to make predictions.

q-bio.PE↗

arXiv2020

A self-supervised neural-analytic method to predict the evolution of COVID-19 in Romania

Analysing and understanding the transmission and evolution of the COVID-19 pandemic is mandatory to be able to design the best social and medical policies, foresee their outcomes and deal with all the subsequent socio-economic effects. We address this important problem from a computational and machine learning perspective. More specifically, we want to statistically estimate all the relevant parameters for the new coronavirus COVID-19, such as the reproduction number, fatality rate or length of infectiousness period, based on Romanian patients, as well as be able to predict future outcomes. This endeavor is important, since it is well known that these factors vary across the globe, and might be dependent on many causes, including social, medical, age and genetic factors. We use a recently published improved version of SEIR, which is the classic, established model for infectious diseases. We want to infer all the parameters of the model, which govern the evolution of the pandemic in Romania, based on the only reliable, true measurement, which is the number of deaths. Once the model parameters are estimated, we are able to predict all the other relevant measures, such as the number of exposed and infectious people. To this end, we propose a self-supervised approach to train a deep convolutional network to guess the correct set of Modified-SEIR model parameters, given the observed number of daily fatalities. Then, we refine the solution with a stochastic coordinate descent approach. We compare our deep learning optimization scheme with the classic grid search approach and show great improvement in both computational time and prediction accuracy. We find an optimistic result in the case fatality rate for Romania which may be around 0.3% and we also demonstrate that our model is able to correctly predict the number of daily fatalities for up to three weeks in the future.

q-bio.PE↗