稀疏性的代價：使用稀疏與稀疏化量測之稀疏恢復的充分條件

摘要

我們考慮從具雜訊線性測量中還原稀疏二元訊號的支撐集問題。對於稀疏高斯測量矩陣，我們辨識出在高訊雜比範圍 ds/p → ∞ 中，最大概似還原所需最小樣本數的充分條件，其中 p 表示訊號維度，s 為訊號非零分量數，d 為測量矩陣每列非零分量的期望數。結合已知下界，這給出階為 s log(p/s) / log(ds/p) 的資訊理論閾值，明確揭示測量稀疏性的代價。特別是，我們指出一個範圍，其中測量稀疏性所造成的樣本複雜度損失為對數級，而計算增益幾乎為線性。其次，我們研究在將原本稠密的高斯設計稀疏化之後的還原：觀測值由該稠密設計生成，而估計使用一個獨立稀疏化的設計以及重新縮放的響應。在比例範圍 s=αp、d=ψp 中，我們證明，對每個固定的目標誤差水準 δ 及每個鬆弛量 ε>0，階為 p/ψ^2 的樣本數即足以在任意小的 ψ 下還原支撐集。

English

We consider the problem of support recovery for sparse binary signals from noisy linear measurements. For sparse Gaussian measurement matrices we identify sufficient conditions on the minimal sample size for maximum-likelihood recovery in the high-SNR regime ds/p to infty, where p denotes the signal dimension, s the number of non-zero components of the signal, and d the expected number of non-zero components per row of measurement. Combined with known lower bounds, this yields an information-theoretic threshold of order slog(p/s) / log(ds/p), making explicit the price of measurement sparsity. In particular, we highlight a regime where the sample-complexity loss from measurement sparsity is logarithmic while the computational gain is nearly linear. Second, we study recovery after sparsifying an originally dense Gaussian design: the observations are generated from the dense design, while estimation uses an independently sparsified design and a rescaled response. In the proportional regime s=αp, d=ψp, we prove that, for every fixed target error level δ and every slack varepsilon>0, a sample size of order p/ψ^2 is sufficient for support recovery for arbitrarily small ψ.