非对称信息编码集
Coding sets with asymmetric information
摘要：
我们研究了以下单向非对称传输问题，也是基于模型的压缩感知的一个变体：资源有限的编码器必须报告一个小集合$S$从宇宙的$N$项目到一个更强大的解码器（服务器）。区别特征是信息不对称：子集$S$由i.i.d.组成。样本来自先验分布$μ$，并且$μ$仅为解码器所知。编码器的目标是对$S$进行编码，同时实现$|S| \cdot H(μ)$的信息理论界，即，香农熵界我们首先表明，任何这样的压缩方案必须是{ \em随机}，如果它从先前的$μ$非平凡的收益。这与对称情况（当编码器和解码器都知道$μ$时）形成对比，在对称情况下，霍夫曼码提供接近最优的确定性解决方案。另一方面，一个相当简单的论点表明，当随机线性码$|S|=k$达到约$k\cdot H(μ)$位的接近最佳的通信速率时。可惜的是，最终方案的解码时间太长：大约${N\choose k} \approx (N/k)^k$。我们的主要结果是一个计算效率和线性编码方案，它实现了$O(\lg\lg N)$竞争力的通信比相比，最佳的基准，并运行在$\text{poly}(N,k)$的时间。我们的“多级”编码方案使用里德-所罗门码的散列和综合征解码的组合，并且依赖于将（未知）先验$μ$视为均匀（“平坦”）分布的相当小的凸组合。
英文摘要：
We study the following one-way asymmetric transmission problem, also a variant of model-based compressed sensing: a resource-limited encoder has to report a small set $S$ from a universe of $N$ items to a more powerful decoder (server). The distinguishing feature is asymmetric information: the subset $S$ is comprised of i.i.d. samples from a prior distribution $μ$, and $μ$ is only known to the decoder. The goal for the encoder is to encode $S$ obliviously, while achieving the information-theoretic bound of $|S| \cdot H(μ)$, i.e., the Shannon entropy bound. We first show that any such compression scheme must be {\em randomized}, if it gains non-trivially from the prior $μ$. This stands in contrast to the symmetric case (when both the encoder and decoder know $μ$), where the Huffman code provides a near-optimal deterministic solution. On the other hand, a rather simple argument shows that, when $|S|=k$, a random linear code achieves near-optimal communication rate of about $k\cdot H(μ)$ bits. Alas, the resulting scheme has prohibitive decoding time: about ${N\choose k} \approx (N/k)^k$. Our main result is a computationally efficient and linear coding scheme, which achieves an $O(\lg\lg N)$-competitive communication ratio compared to the optimal benchmark, and runs in $\text{poly}(N,k)$ time. Our "multi-level" coding scheme uses a combination of hashing and syndrome-decoding of Reed-Solomon codes, and relies on viewing the (unknown) prior $μ$ as a rather small convex combination of uniform ("flat") distributions.
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