arXiv subjects

Chandan Pradhan

Publications and source records attributed to Chandan Pradhan.

At least 19 recordsLinked to original sources

arXiv2026

Finite sum of squares, finite realization and noncommutative Carath\'eodory approximation

In the noncommutative polydisc, we first prove a positive sum of squares formula for a non-negative hereditary rational nc-function. The number of summands is finite. This result is used to derive a finite-dimensional realization formula for contractive nc-rational functions, where the colligation matrix is contractive. It is unitary if and only if the function is inner. Finally, we apply these results to generalize Carath\'eodory's classical theorem - approximating holomorphic self-maps of the unit disc by finite Blaschke products - to the setting of holomorphic functions on the noncommutative polydisc. This is in sharp contrast with the commutative situation where Carath\'eodory's approximation is known for Schur classes only in the unit disc and the unit bidisc.

math.FA↗

arXiv2026

Z-Opt: A Near-Optimal Reduced-Complexity Two-Dimensional Grassmannian Constellation

Grassmannian constellations are known to achieve the capacity of noncoherent communications over Rayleigh fading channels in the high-SNR regime, yet their efficient construction remains challenging. In this paper, we propose two construction methods for Grassmannian constellations of one-dimensional subspaces in a two-dimensional space, termed S-Opt and Z-Opt, along with two low-complexity detectors. Both the construction and detection procedures are performed on the unit sphere, known as the Bloch sphere in quantum computing. We show that the chordal distance on the Grassmann manifold is proportional to the Euclidean distance on the Bloch sphere and derive a corresponding theoretical upper bound based on the Fejes--T\'oth bound on the minimum chordal distance. The S-Opt constellation is constructed from sphere-packing solutions and attains the derived upper bound for the optimal Bloch-sphere packings considered. The S-Opt detector can be applied to arbitrary Grassmannian constellations on $\mathcal{G}(2,1)$, and its time complexity scales linearly with the number of receive antennas and logarithmically with the constellation size, while yielding the same detection performance as the GLRT detector. Furthermore, based on the insight obtained through the S-Opt construction, the Z-Opt constellation is constructed by stacking regular polygons on the Bloch sphere, and its minimum chordal distance approaches the derived upper bound over the evaluated constellation sizes. The Z-Opt detector's time complexity scales linearly with the number of receive antennas, while yielding the same detection performance as the GLRT detector for Z-Opt.

cs.IT↗

arXiv2026

Isometric Embeddability of Schatten Classes Revisited

In this note, we summarize known results and open questions on the existence of isometric embeddings between different Schatten classes as well as obtain a new non-embeddability result using a novel method. We also provide a brief overview of the relevant methods.

math.FA↗

arXiv2026

Low-Complexity and Power-Efficient Precoding Codebook Design on Sparse Grassmannian

We propose a sparse Grassmannian design for precoding codebooks. Due to their sparse structure, our proposed codebooks achieve low peak-to-average power ratio (PAPR), low complexity of precoder multiplication, and low storage cost, while demonstrating performance comparable to the optimal codebook. Specifically, we introduce a method for constructing codebooks based on Schubert cell decomposition on the Grassmann manifold. Designing an optimal Grassmannian precoding codebook generally requires high computational complexity. In the proposed approach, by exploiting its sparsity, the objective function can be simplified, and the search space can also be significantly reduced compared to state-of-the-art codebooks. Numerical simulations in uplink systems demonstrate that the proposed sparse codebook asymptotically approaches the optimal codebook and outperforms the codebook currently adopted in 5G NR, in terms of achievable rate under uncorrelated Rayleigh fading channels, while maintaining substantially lower PAPR than conventional dense designs. These results confirm that the proposed sparse codebook can be a practical and power-efficient alternative to conventional codebooks for a wide range of uplink transmission scenarios.

eess.SP↗

arXiv2026

Noncommutative $L_p$-differentiability and trace formulae

Let $\mathcal{M}$ be a semifinite von Neumann algebra equipped with a normal faithful semifinite trace $\tau$, and let $L_p(\mathcal{M})$ denote the associated noncommutative $L_p$-space for $1<p<\infty$. Let $n\in\mathbb{N}$ and let $a, b$ be $\tau$-measurable self-adjoint operators such that $b\in L_p(\mathcal{M})\cap L_{np}(\mathcal{M})$. For a function $f\in C^n(\mathbb{R})$ whose derivatives $f^{(k)}$ are bounded for $1\le k\le n$, we prove that the map $\phi:t\in\mathbb{R}\mapsto f(a+tb)-f(a)$ is $n$-times differentiable in the $\|\cdot\|_{L_p}$-norm. This strengthens the corresponding result of de Pagter and Sukochev for $p\neq 2$ and extends it to higher-order derivatives. In addition, if $f^{(n)}\in C_0(\mathbb{R})$ or $b\in \mathcal{M}$, then $\phi^{(n)}$ is continuous on $\mathbb{R}$. Consequently, we extend the Potapov--Skripka--Sukochev higher-order trace formula from bounded $L_n$-perturbations to not necessarily bounded perturbations in $L_n(\mathcal{M})\cap L_{n^{2}}(\mathcal{M})$. Moreover, we show that this trace formula holds for a broader class of admissible functions than the classes previously considered in the literature.

math.OA↗

arXiv2026

Sparse Grassmannian Design for Noncoherent Codes via Schubert Cell Decomposition

In this paper, we propose a method for designing sparse Grassmannian codes for noncoherent multiple-input multiple-output systems. Conventional pairwise error probability formulations under uncorrelated Rayleigh fading channels fail to account for rank deficiency induced by sparse configurations. We revise these formulations to handle such cases in a unified manner. Furthermore, we derive a closed-form metric that effectively maximizes the noncoherent average mutual information (AMI) at a given signal-to-noise ratio. We focus on the fact that the Schubert cell decomposition of the Grassmann manifold provides a mathematically sparse property, and establish design criteria for sparse noncoherent codes based on our analyses. In numerical results, the proposed sparse noncoherent codes outperform conventional methods in terms of both symbol error rate and AMI, and asymptotically approach the performance of the optimal Grassmannian constellations in the high-signal-to-noise ratio regime. Moreover, they reduce the time and space complexity, which does not scale with the number of transmit antennas.

eess.SP↗

arXiv2025

Positivity of spectral shift functions and infinite-dimensional BMV conjecture

We obtain a solution to the Bessis-Moussa-Villani conjecture for a trace-class perturbation of a semi-bounded operator and answer affirmatively the question on positivity of higher order spectral shift functions in the setting of Schatten--von Neumann perturbations of (possibly unbounded) self-adjoint operators.

math.FA↗

arXiv2025

Maximizing Spectrum Efficiency of Data-Carrying Reference Signals via Bayesian Optimization

Data-carrying reference signals are a type of reference signal (RS) constructed on the Grassmann manifold, which allows for simultaneous data transmission and channel estimation to achieve boosted spectral efficiency at high signal-to-noise ratios (SNRs). However, they do not improve spectral efficiency at low to middle SNRs compared with conventional RSs. To address this problem, we propose a numerical optimization-based Grassmann constellation design on the Grassmann manifold that accounts for both data transmission and channel estimation. In our numerical optimization, we derive an upper bound on the normalized mean squared error (NMSE) of estimated channel matrices and a lower bound on the noncoherent average mutual information (AMI), and these bounds are optimized simultaneously by using a Bayesian optimization technique. The proposed objective function outperforms conventional design metrics in obtaining Pareto-optimal constellations for NMSE and AMI. The constellation obtained by our method achieves an NMSE comparable to conventional non-data-carrying RSs while enabling data transmission, resulting in superior AMI performance and improved spectral efficiency even at middle SNRs.

eess.SP↗

arXiv2025

Superimposed Pilot-Based OTFS: Will It Work?

Orthogonal time frequency space (OTFS) modulation is a promising solution to handle doubly-selective fading, but its channel estimation is a nontrivial task in terms of maximizing spectral efficiency. Conventional pilot assignment approaches face challenges: the standard embedded pilot-based scheme suffers from low transmission rates, and the single superimposed pilot (SP)-based scheme experiences inevitable data-pilot interference, leading to coarse channel estimation. To cope with this issue, focusing on the SP-based OTFS system in channel coded scenarios, we propose a novel pilot assignment scheme and an iterative algorithm. The proposed scheme allocates multiple SPs per frame to estimate channel coefficients accurately. Furthermore, the proposed algorithm performs refined interference cancellation, utilizing a replica of data symbols generated from soft-decision outputs provided by a decoder. Assuming fair and unified conditions, we evaluate each pilot assignment scheme in terms of reliability, channel estimation accuracy, effective throughput, and computational complexity. Our numerical simulations demonstrate that the multiple SP-based scheme, which balances the transmission rate and the interference cancellation performance, has the best throughput at the expense of slightly increased complexity. In addition, we confirm that the multiple SP-based scheme achieves further improved throughput due to the proposed interference cancellation algorithm.

eess.SP↗

arXiv2024

Trace formulas for $\mathcal{S}^p$-perturbations and extension of Koplienko-Neidhardt trace formulas

In this paper, we extend the class of admissible functions for the trace formula of the second order in the self-adjoint, unitary, and contraction cases for a perturbation in the Hilbert-Schmidt class $\mathcal{S}^2(\mathcal{H})$ by assuming a certain factorization of the divided difference $f^{[2]}$. This class is the natural one to ensure that the second order Taylor remainder is a trace class operator. It encompasses all the classes of functions for which the trace formula was previously known. Secondly, for a Schatten $\mathcal{S}^p$-perturbation, $1<p<\infty$, we prove general modified trace formulas for every $n$-times differentiable functions with bounded $n$-th derivative in the self-adjoint and unitary cases and for every $f$ such that $f$ and its derivatives are in the disk algebra $\mathcal{A}(\mathbb{D})$ in the contraction case.

math.FA↗

arXiv2024

Gromov-Hausdorff convergence of metric spaces of UCP maps

It is shown that van Suijlekom's technique of imposing a set of conditions on operator system spectral triples ensures Gromov-Hausdorff convergence of sequences of sets of unital completely positive maps (equipped with the BW-topology which is metrizable). This implies that even when only a part of the spectrum of the Dirac operator is available together with a certain truncation of the $C^*$-algebra, information about the geometry can be extracted.

math.OA↗

arXiv2024

Covert Communications Without Pre-Sharing of Side Information and Channel Estimation Over Quasi-Static Fading Channels

We propose a new covert communication scheme that operates without pre-sharing side information and channel estimation, utilizing a Gaussian-distributed Grassmann constellation for noncoherent detection. By designing constant-amplitude symbols on the Grassmann manifold and multiplying them by random variables, we generate signals that follow an arbitrary probability distribution, such as Gaussian or skew-normal distributions. The mathematical property of the manifold enables the transmitter's random variables to remain unshared with the receiver, and the elimination of pilot symbols that could compromise covertness. The proposed scheme achieved higher covertness and achievable rates compared to conventional coherent Gaussian signaling schemes, without any penalty in terms of complexity.

eess.SP↗

arXiv2024

Higher-Order Trace Formulas for Contractive and Dissipative Operators

We establish higher order trace formulas for pairs of contractions along a multiplicative path generated by a self-adjoint operator in a Schatten-von Neumann ideal, removing earlier stringent restrictions on the kernel and defect operator of the contractions and enlarging the set of admissible functions. We also derive higher order trace formulas for maximal dissipative operators under relaxed assumptions and new simplified trace formulas for unitary and resolvent comparable self-adjoint operators. The respective spectral shift measures are absolutely continuous and, in the case of contractions, the set of admissible functions for the $n$th order trace formula on the unit circle includes the Besov class $B^n_{\infty, 1}(\T)$. Both aforementioned properties are new in the mentioned generality.

math.FA↗

arXiv2024

Differentiation, Taylor series, and all order spectral shift functions, for relatively bounded perturbations

Given $H$ self-adjoint, $V$ symmetric and relatively $H$-bounded, and $f:\mathbb{R}\to\mathbb{C}$ satisfying mild conditions, we show that the Gateaux derivative $$\frac{d^n}{dt^n}f(H+tV)|_{t=0}$$ exists in the operator norm topology, for every natural $n$, give a new explicit formula for this derivative in terms of multiple operator integrals, and establish useful perturbation formulas for multiple operator integrals under relatively bounded perturbations. Moreover, if the $H$-bound of $V$ is less than 1, we obtain sufficient conditions on $f$ which ensure that the Taylor expansion $$f(H+V)=\sum_{n=0}^\infty\frac{1}{n!}\frac{d^n}{dt^n} f(H+tV)\big|_{t=0}$$ exists and converges absolutely in operator norm. Finally, assuming that $V(H-i)^{-p}\in\mathcal{S}^{s/p}$ for $p=1,\ldots,s$ for some $s\in\mathbb{N}$ (for instance, when $H$ is an order 1 differential operator on an $s-1$ dimensional space), we show that the Krein--Koplienko spectral shift functions $\eta_{k,H,V}$, satisfying $${Tr}\left(f(H+V)-\sum_{m=0}^{k-1}\frac{1}{m!}\frac{d^m}{dt^m} f(H+tV)\big|_{t=0}\right)=\int_{\mathbb{R}} f^{(k)}(x)\eta_{k,H,V}(x)dx,$$ exist for every $k=1,2,3,\ldots$, independently of $s$. The latter result (which is significantly stronger than \cite{vNS22}) is completely new also in the case that $V$ is bounded. The proof is based on \cite{PSS}, combined with a generalisation of the multiple operator integral compatible with \cite{HMvN}. We discuss applications of our results to quantum physics and noncommutative geometry.

math.FA↗

arXiv2024

Higher order $\mathcal{S}^{p}$-differentiability: The unitary case

Consider the set of unitary operators on a complex separable Hilbert space $\hilh$, denoted as $\mathcal{U}(\hilh)$. Consider $1<p<\infty$. We establish that a function $f$ defined on the unit circle $\cir$ is $n$ times continuously Fr\'echet $\Sp^p$-differentiable at every point in $\mathcal{U}(\hilh)$ if and only if $f\in C^n(\cir)$. Take a function $U :\R\rightarrow\mathcal{U}(\hilh)$ such that the function $t\in\R\mapsto U(t)-U(0)$ takes values in $\Sp^{p}$ and is $n$ times continuously $\Sp^{p}$-differentiable on $\R$. Consequently, for $f\in C^n(\cir)$, we prove that $f$ is $n$ times continuously G\^ateaux $\mathcal{S}^p$-differentiable at $U(t)$. We provide explicit expressions for both types of derivatives of $f$ in terms of multiple operator integrals. In the domain of unitary operators, these results closely follow the $n$th order successes for self-adjoint operators achieved by the second author, Le Merdy, Skripka, and Sukochev. Furthermore, as for application, we derive a formula and $\Sp^{p}$-estimates for operator Taylor remainders for a broader class of functions. Our results extend those of Peller, Potapov, Skripka, Sukochev and Tomskova.

math.FA↗

arXiv2024

Boosting Spectral Efficiency with Data-Carrying Reference Signals on the Grassmann Manifold

In wireless networks, frequent reference signal transmission for accurate channel reconstruction may reduce spectral efficiency. To address this issue, we consider to use a data-carrying reference signal (DC-RS) that can simultaneously estimate channel coefficients and transmit data symbols. Here, symbols on the Grassmann manifold are exploited to carry additional data and to assist in channel estimation. Unlike conventional studies, we analyze the channel estimation errors induced by DC-RS and propose an optimization method that improves the channel estimation accuracy without performance penalty. Then, we derive the achievable rate of noncoherent Grassmann constellation assuming discrete inputs in multi-antenna scenarios, as well as that of coherent signaling assuming channel estimation errors modeled by the Gauss-Markov uncertainty. These derivations enable performance evaluation when introducing DC-RS, and suggest excellent potential for boosting spectral efficiency, where interesting crossings with the non-data carrying RS occurred at intermediate signal-to-noise ratios.

eess.SP↗

arXiv2023

Lipschitz Estimates and an application to trace formulae

In this note, we provide an elementary proof for the expression of $f(U)-f(V)$ in the form of a double operator integral for every Lipschitz function $f$ on the unit circle $\cir$ and for a pair of unitary operators $(U,V)$ with $U-V\in\mathcal{S}_{2}(\hilh)$ (the Hilbert-Schmidt class). As a consequence, we obtain the Schatten $2$-Lipschitz estimate $\|f(U)-f(V)\|_2\leq \|f\|_{\lip(\cir)}\|U-V\|_2$ for all Lipschitz functions $f:\cir\to\C$. Moreover, we develop an approach to the operator Lipschitz estimate for a pair of contractions with the assumption that one of them is a strict contraction, which significantly extends the class of functions from results known earlier. More specifically, for each $p\in(1,\infty)$ and for every pair of contractions $(T_0,T_1)$ with $\|T_0\|<1$, there exists a constant $d_{f, p,T_0}>0$ such that $\|f(T_1)-f(T_0)\|_p\leq d_{f,p, T_0}\|T_1-T_0\|_p$ for all Lipschitz functions on $\cir$. Using our Lipschitz estimates, we establish a modified Krein trace formula applicable to a specific category of pairs of contractions featuring Hilbert-Schmidt perturbations.

math.FA↗

arXiv2023

Estimates and Higher-Order Spectral Shift Measures in Several Variables

In recent years, higher-order trace formulas of operator functions have attracted considerable attention to a large part of the perturbation theory community. In this direction, we prove estimates for traces of higher-order derivatives of multivariable operator functions with associated scalar functions arising from multivariable analytic function space and, as a consequence, derive higher-order spectral shift measures for pairs of tuples of commuting contractions under Hilbert-Schmidt perturbations. These results substantially extend the main results of \cite{Sk15}, where the estimates were proved for traces of first and second-order derivatives of multivariable operator functions. In the context of the existence of higher-order spectral shift measures, our results extend the relative results of \cite{DySk09, PoSkSu14} from a single-variable to a multivariable setting under Hilbert-Schmidt perturbations. Our results rely crucially on heavy uses of explicit expressions of higher-order derivatives of operator functions and estimates of the divided deference of multivariable analytic functions, which are developed in this paper, along with the spectral theorem of tuples of commuting normal operators.

math.FA↗