On the Fundamental Limits of Recovering Tree Sparse Vectors From Noisy Linear Measurements

Abstract

Recent breakthrough results in compressive sensing (CS) have established that many high dimensional signals can be accurately recovered from a relatively small number of non-adaptive linear observations, provided that the signals possess a sparse representation in some basis. Subsequent efforts have shown that the performance of CS can be improved by exploiting additional structure in the locations of the nonzero signal coefficients during inference, or by utilizing some form of data-dependent adaptive measurement focusing during the sensing process. To our knowledge, our own previous work was the first to establish the potential benefits that can be achieved when fusing the notions of adaptive sensing and structured sparsity -that work examined the task of support recovery from noisy linear measurements, and established that an adaptive sensing strategy specifically tailored to signals that are tree-sparse can significantly outperform adaptive and non-adaptive sensing strategies that are agnostic to the underlying structure.In this work we establish fundamental performance limits for the task of support recovery of tree-sparse signals from noisy measurements, in settings where measurements may be obtained either non-adaptively (using a randomized Gaussian measurement strategy motivated by initial CS investigations) or by any adaptive sensing strategy. Our main results here imply that the adaptive tree sensing procedure analyzed in our previous work is nearly optimal, in the sense that no other sensing and estimation strategy can perform fundamentally better for identifying the support of tree-sparse signals. Index TermsAdaptive sensing, compressive sensing, minimax lower bounds, sparse support recovery, structured sparsity, tree sparsity.

Recovering graph-structured activations using adaptive compressive measurements

“…al. give necessary and sufficient conditions for recovering a small square of activiations in a grid [9] while Soni and Haupt analyze the recovery of treesparse signals [7], [12]. Our work provides guarantees that depends on how well the signal is captured by our algorithmic construction.…”

Fundamental limits for support recovery of tree-sparse signals from noisy compressive samples

“…We evaluate four different sensing and support estimation strategies -a nonadaptive CS strategy based on the Lasso estimator; non-adaptive CS using a group-Lasso estimator, with groups designed to enforce tree-structure; the adaptive CS procedure of [23]; and the adaptive tree sensing procedure described above -intended to be illustrative approaches for each of the four scenarios identified in Table I. We refer reader to [22] for complete details on the experimental setup.…”

“…We state the results here as theorems, and refer the readers to the full version of this paper [22] for detailed proofs. Our first main result analyzes the support recovery task for tree-sparse signals in a non-adaptive sensing scenario.…”

On the effect of model mismatch for sequential Info-Greedy Sensing

“…We may classify these algorithms as (1) being agnostic about the signal distribution and, hence, random measurements are used [7][8][9][10], (2) exploiting additional structure of the signal (such as graphical structure [11], sparse [12][13][14], low rank [15], and treesparse structure [16,17]) to design measurements, and (3) exploiting the distributional information of the signal in choosing measurements [18], possibly through maximizing mutual information. The additional knowledge about signal structure or distributions are various forms of information about the unknown signal.…”