Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1502.02245

Restricted Isometry Property of Subspace Projection Matrix Under Random Compression

Xinyue Shen·Yuantao Gu
View original source ↗Read source document ↗Export metadata ↓

Abstract

Structures play a significant role in the field of signal processing. As a representative of structural data, low rank matrix along with its restricted isometry property (RIP) has been an important research topic in compressive signal processing. Subspace projection matrix is a kind of low rank matrix with additional structure, which allows for further reduction of its intrinsic dimension. This leaves room for improving its own RIP, which could work as the foundation of compressed subspace projection matrix recovery. In this work, we study the RIP of subspace projection matrix under random orthonormal compression. Considering the fact that subspace projection matrices of $s$ dimensional subspaces in $\mathbb{R}^N$ form an $s(N-s)$ dimensional submanifold in $\mathbb{R}^{N\times N}$, our main concern is transformed to the stable embedding of such submanifold into $\mathbb{R}^{N\times N}$. The result is that by $O(s(N-s)\log N)$ number of random measurements the RIP of subspace projection matrix is guaranteed.

Explore related subjects

cs.ITmath.IT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Xinyue Shen, Yuantao Gu. 2015-02-08. Restricted Isometry Property of Subspace Projection Matrix Under Random Compression. https://doi.org/10.1109/lsp.2015.2402206

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the Distribution of Age of Information in Time-varying Updating Systems

Age of Information (AoI) is a crucial metric for quantifying information freshness in real-time systems where the sampling rate of data packets is time-varying. Evaluating AoI under such conditions is challenging, as system states become temporally correlated and traditional stationary analysis is inapplicable. We investigate an $M_{t}/G/1/1$ queueing system with a time-varying sampling rate and probabilistic preemption, proposing a novel analytical framework based on multi-dimensional partial differential equations (PDEs) to capture the time evolution of the system's status distribution. To solve the PDEs, we develop a decomposition technique that breaks the high-dimensional PDE into lower-dimensional subsystems. Solving these subsystems allows us to derive the Aol distribution at arbitrary time instances. We show AoI does not exhibit a memoryless property, even with negligible processing times, due to its dependence on the historical sampling process. Our framework extends to the stationary setting, where we derive a closed-form expression for the Laplace-Stieltjes Transform (LST) of the steady-state AoI. Numerical experiments reveal AoI exhibits a non-trivial lag in response to sampling rate changes. Our results also show that no single preemption probability or processing time distribution can minimize Aol violation probability across all thresholds in either time-varying or stationary scenarios. Finally, we formulate an optimization problem and propose a heuristic method to find sampling rates that reduce costs while satisfying AoI constraints.

cs.IT↗
arXiv2026

On the hull of linearized polynomial codes

Motivated by entanglement-assisted quantum error-correcting codes, where the hull dimension determines the number of required pre-shared entangled pairs, we study hulls of two families of $\mathbb{F}_q$-linear codes defined by $q$-polynomial operators over $\mathbb{F}_{q^m}$. Our main tool is a unified Gram-matrix method. For image codes $\mathcal{C}(\boldsymbolα)=\operatorname{im}Φ_{\boldsymbolα}$, with $Φ_{\boldsymbolα}=\sum_iα_iF_i$, we prove the master hull--rank formula $\dim\operatorname{Hull}(\mathcal{C}(\boldsymbolα))=\operatorname{rank}(Φ_{\boldsymbolα})-\operatorname{rank}(G(\boldsymbolα))$, where $G(\boldsymbolα)$ is the associated Gram matrix over $\mathbb{F}_q$. Specializing to $C_{λ,μ}=\operatorname{im}(λx+μL(x))$, we obtain a quadratic Gram pencil $λ^2G_0+λμG_1+μ^2G_2$ whose determinant describes the LCD locus in $\mathbb{P}^1(\mathbb{F}_q)$. We also treat $\mathbb{F}_{q^m}$-linear rank-distance codes $\mathcal{C}=\langle X,F_1,\ldots,F_k\rangle_{\mathbb{F}_{q^m}}$ with the Delsarte inner product, where a $k\times k$ Gram matrix over $\mathbb{F}_{q^m}$ determines the hull dimension. For $L(X)=X^{q^k}$, with $d=\gcd(k,m)$, the resulting circulant Gram matrices yield a closed-form discriminant and a complete classification in three of the four bijectivity configurations over $\mathbb{P}^1(\mathbb{F}_{q^m})$. In the remaining case, the hull dimension equals $δ=\dim_{\mathbb{F}_q}(\operatorname{im}ϕ_{λ,μ}\cap\kerϕ_{λ,μ}^{\dagger})$, and the extremal condition $δ=d$ is characterized by an explicit trace-isotropy criterion. We conclude with an exact count of LCD and non-LCD points, showing that the LCD density tends to $1$ as $q\to\infty$, together with a worked example over $\mathbb{F}_{64}$ and a SageMath verification.

cs.IT↗
arXiv2026

Low-Complexity Run-Length-Limited ISI-Mitigation (RLIM) Codes for Molecular Communication

Molecular communication suffers from severe inter-symbol interference, which makes constrained coding essential for reliable transmission. Run-length-limited ISI-mitigation codes are attractive because they select low-weight constrained codebooks, reducing ISI while allowing more molecules to be assigned to each transmitted \(1\)-symbol under the usual molecular-communication normalization. Previous results showed strong bit-error-rate performance for these codes, but their original realization required full codebook generation and storage. This exponential storage growth is unsuitable for resource-constrained molecular communication channels and also limits the exploration of larger information dimensions. This is particularly important for nano-scale molecular communication, where transmitter and receiver nodes are expected to operate under severe memory and computational constraints. This paper removes that realization bottleneck by replacing full codebook storage with an enumerative realization based on Cover's ranking framework, constant-weight run-length-limited counting, and cumulative weight-layer offsets. The resulting encoder and decoder preserve the selected RLIM codebooks and the original projection-based decoding behavior while storing only polynomial-size counting tables. Storage and runtime measurements confirm the resulting exponential-to-polynomial reduction, while diffusion-based molecular-communication simulations show that the newly accessible larger information dimensions can improve the best mean bit-error-rate performance attainable by individual RLIM orders in several tested channel regimes.

cs.IT↗