Random subspaces $X$ of $\mathbb{R}^n$ of dimension proportional to $n$ are, with high probability, well-spread with respect to the $\ell_p$-norm (for $p \in [1,2]$). Namely, every nonzero $x \in X$ is "robustly non-sparse" in the following sense: $x$ is $\varepsilon \|x\|_p$-far in $\ell_p$-distance from all $\delta n$-sparse vectors, for positive constants $\varepsilon, \delta$ bounded away from $0$. This "$\ell_p$-spread" property is the natural counterpart, for subspaces over the reals, of the minimum distance of linear codes over finite fields, and, for $p = 2$, corresponds to $X$ being a Euclidean section of the $\ell_1$ unit ball. Explicit $\ell_p$-spread subspaces of dimension $\Omega(n)$, however, are not known except for $p=1$. The construction for $p=1$, as well as the best known constructions for $p \in (1,2]$ (which achieve weaker spread properties), are analogs of low density parity check (LDPC) codes over the reals, i.e., they are kernels of sparse matrices. We study the spread properties of the kernels of sparse random matrices. Rather surprisingly, we prove that with high probability such subspaces contain vectors $x$ that are $o(1)\cdot \|x\|_2$-close to $o(n)$-sparse with respect to the $\ell_2$-norm, and in particular are not $\ell_2$-spread. On the other hand, for $p < 2$ we prove that such subspaces are $\ell_p$-spread with high probability. Moreover, we show that a random sparse matrix has the stronger restricted isometry property (RIP) with respect to the $\ell_p$ norm, and this follows solely from the unique expansion of a random biregular graph, yielding a somewhat unexpected generalization of a similar result for the $\ell_1$ norm [BGI+08]. Instantiating this with explicit expanders, we obtain the first explicit constructions of $\ell_p$-spread subspaces and $\ell_p$-RIP matrices for $1 \leq p < p_0$, where $1 < p_0 < 2$ is an absolute constant.

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$\ell_p$-稀疏矩阵的扩展属性

与 $n$ 成正比的 $\mathbb{R}^n$ 的随机子空间 $X$ 很有可能相对于 $\ell_p$-范数（对于 $p \in [1, 2]$）。即，每个非零 $x \in X$ 在以下意义上都是“稳健非稀疏的”： $x$ 是 $\varepsilon \|x\|_p$-far in $\ell_p$-距所有 $\delta n$-稀疏向量，对于正常数 $\varepsilon，\delta$ 有界于 $0$。这个 "$\ell_p$-spread" 属性是实数上的子空间、有限域上线性代码的最小距离的自然对应物，并且对于 $p = 2$，对应于 $X$ 是欧几里得截面$\ell_1$ 单位球。然而，维度 $\Omega(n)$ 的显式 $\ell_p$-spread 子空间除了 $p=1$ 之外是未知的。$p=1$ 的构造，以及 $p \in (1, 2]$（实现较弱的扩展特性）是实数上的低密度奇偶校验（LDPC）码的类似物，即它们是稀疏矩阵的核。我们研究稀疏随机矩阵核的传播特性。相当令人惊讶的是，我们证明这样的子空间很有可能包含 $o(1)\cdot \|x\|_2$-接近 $o(n)$-相对于 $\ell_2 稀疏的向量 $x$ $-norm，特别是不是 $\ell_2$-spread。另一方面，对于 $p < 2$，我们证明这样的子空间是 $\ell_p$-spread 的概率很高。此外，我们表明随机稀疏矩阵相对于 $\ell_p$ 范数具有更强的受限等距属性（RIP），这完全来自随机双正则图的唯一扩展，对 $\ell_1$ 范数 [BGI+08] 的类似结果产生了一些出乎意料的概括。使用显式扩展器对其进行实例化，我们获得 $\ell_p$-spread 子空间和 $\ell_p$-RIP 矩阵的第一个显式构造，用于 $1 \leq p < p_0$，其中 $1 < p_0 < 2$ 是绝对常数。