arXiv subjects

Massoud Heidari

Publications and source records attributed to Massoud Heidari.

4 recordsLinked to original sources

arXiv2026

Deterministic Causal Quantization with Bounded Error: Adaptive Reconstruction and Entropy bounds

We study compression of real-valued stationary sources under a deterministic error bound: every sample must be reconstructed within a prescribed tolerance $\varepsilon$ on every input sequence. Reconstructions are causal, depending only on present and past inputs. Rates are measured by the entropy of the quantizer output, which block entropy coding can approach, and compared with the minimum rate of unrestricted block quantization under the same error constraint. A uniform scalar quantizer with cells of width $2\varepsilon$ is within one bit per sample of this minimum whenever finite-entropy scalar quantization is possible, without density, independence, or mixing assumptions. Under entropy and Sobolev regularity conditions, the gap is at most $2\mathcal{J}\varepsilon^2/(π^2\ln 2)$, where $\mathcal{J}$ is the asymptotic Fisher information per sample. Thus the cost of causal reconstruction vanishes at a quadratic rate as the tolerance decreases. A uniform grid, however, cannot adapt to the source. We construct an adaptive quantizer whose encoder and decoder maintain a common finite set of reconstruction values, updated from the transmitted symbols alone; suitable initialization reproduces any uniform grid. When its initial values are at least $\varepsilon$ apart, its rate is within two bits per sample of the minimum for every such source. For sources uniform within separated intervals of width $2\varepsilon$, storing the interval centers attains the minimum and its Shannon lower bound. Such configurations can outperform every uniform grid of the same cell width: over sources with well-separated intervals, the best uniform grid's worst-case excess rate is $1/(2\ln 2)\approx 0.72$ bit per sample.

cs.IT↗

arXiv2026

Optimal rounding under integer constraints

Given $N$ real numbers whose sum is an integer, we study the problem of finding $N$ integers that preserve the sum while minimizing the rounding error. We first show that every optimal solution necessarily rounds each coordinate either to its floor or its ceiling, reducing the problem to the selection of the coordinates to be rounded upward. This characterization extends to a class of separable convex integer optimization problems with a single sum constraint. For the resulting optimization problem we characterize the complete set of optimal solutions and show that rounding upward the largest fractional parts simultaneously minimizes the $L^q$ rounding error for every $q\geq 1$. More generally, the resulting error vector is minimal in the weak-majorization order and therefore minimizes every symmetric convex coordinatewise nondecreasing loss of the rounding error. When the $L^q$-optimal solution is not unique, we provide an explicit tie-breaking rule that minimizes the relative rounding error among all optimal solutions. These results lead to a deterministic algorithm with linear $O(N)$ worst-case complexity. Unlike independent randomized rounding, which preserves the target coordinates and the sum constraint only in expectation, the proposed method computes an exactly feasible, provably optimal integer rounding with deterministic optimality guarantee. Besides solving the constrained rounding problem, the algorithm applies as the rounding step in relaxed integer optimization problems with a single conservation constraint.

cs.DS↗

arXiv2022

Post trade allocation: how much are bunched orders costing your performance?

Individual trade orders are often bunched into a block order for processing efficiency, where in post execution, they are allocated into individual accounts. Since Regulators have not mandated any specific post trade allocation practice or methodology, entities try to rigorously follow internal policies and procedures to meet the minimum Regulatory ask of being procedurally fair and equitable. However, as many have found over the years, there is no simple solution for post trade allocation between accounts that results in a uniform distribution of returns. Furthermore, in many instances, the divergences between returns do not dissipate with more transactions, and tend to increase in some cases. This paper is the first systematic treatment of trade allocation risk. We shed light on the reasons for return divergence among accounts, and we present a solution that supports uniform allocation of return irrespective of number of accounts and trade sizes.

q-fin.TR↗

arXiv2015

A Statistical Theory of Deep Learning via Proximal Splitting

In this paper we develop a statistical theory and an implementation of deep learning models. We show that an elegant variable splitting scheme for the alternating direction method of multipliers optimises a deep learning objective. We allow for non-smooth non-convex regularisation penalties to induce sparsity in parameter weights. We provide a link between traditional shallow layer statistical models such as principal component and sliced inverse regression and deep layer models. We also define the degrees of freedom of a deep learning predictor and a predictive MSE criteria to perform model selection for comparing architecture designs. We focus on deep multiclass logistic learning although our methods apply more generally. Our results suggest an interesting and previously under-exploited relationship between deep learning and proximal splitting techniques. To illustrate our methodology, we provide a multi-class logit classification analysis of Fisher's Iris data where we illustrate the convergence of our algorithm. Finally, we conclude with directions for future research.

stat.ML↗