发表机构
School of Engineering, The University of British Columbia, Okanagan Campus; Department of Electrical and Electronics Engineering, Bilkent University（不列颠哥伦比亚大学工程学院; 比尔肯特大学电气与电子工程系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
通过将游程分布视为自由参数并用学习优化，改进了二进制删除信道容量的下界，在多个删除概率上超过现有记录，最大绝对增益3.7e-3比特。
AI 中文摘要
独立同分布二进制删除信道的容量 $C(d)$ 由Dobrushin信息稳定性定理保证存在，但尚无闭式解。经典的构造性下界来自独立同分布游程编码，且仅针对一参数或两参数族（几何、马尔可夫或莫尔斯型）进行了评估。我们证明，当游程分布 $P$ 被视为自由分布并通过学习进行优化时，相同的无限块长泛函会变得严格更强。我们将Drinea--Mitzenmacher泛函简化为关于 $P$ 的双线性形式，并证明有限支撑截断是单侧的，因此计算值仍为有效的下界。我们将Venkataramanan等人的约简从几何游程扩展到任意有限支撑分布，包括用于输出比特熵的残差游程隐马尔可夫模型。Softmax梯度上升搜索 $P$；每个报告的数字都是公式的一次全新单侧评估，无蒙特卡洛，也无有限长度熵惩罚。两个优化下界的包络在每一个测试的 $d$ 值上均超过Gallager的 $1-h(d)$（对于 $d<1/2$）以及Drinea--Mitzenmacher、Venkataramanan等人和Rubinstein--Con的表格化下界。代表性数值：在 $d=0.01$, $0.05$, $0.10$, $0.20$, $0.30$, $0.50$, $0.80$, $0.90$ 时，$C(d)\ge 0.92212$, $0.72939$, $0.56486$, $0.35127$, $0.22616$, $0.10414$, $0.02891$, $0.01322$。相对于该记录的最大绝对增益为 $3.7\times 10^{-3}$ 比特（在 $d=0.30$ 处）；最大相对增益为 $6.8\\%$（在 $d=0.90$ 处）。对于 $d\le 0.45$，包络为自由-$P$ Venkataramanan泛函；从 $d=0.50$ 起为学习到的Drinea--Mitzenmacher分布。在大 $d$ 值下，优化器找到参数族无法表示的稀疏游程梳。Papailiopoulos的一个并发包络在 $[0,1]$ 的大部分区域更强，但我们的包络在高 $d$ 值下仍然更大（例如，在 $d=0.80$ 时为 $0.02891$ 对比 $0.02884$；在 $d=0.90$ 时为 $0.01322$ 对比 $0.01293$）。
英文摘要
The best constructive lower bounds on the capacity of the binary deletion channel come from random codes with independent run lengths, yet the two strongest such bounds, due to Drinea and Mitzenmacher and to Venkataramanan, Tatikonda, and Ramchandran, have been evaluated mainly for geometric or low-parameter run-length laws. We let a learning algorithm choose the run-length law freely, which raises three challenges. First, the Drinea-Mitzenmacher functional is an infinite sum over the ways deletions merge runs; we show that it depends on the law only through its mean, its full-deletion probability, and bilinear forms in the law and its renewal weights, so that gradients of truncations are exact and every truncation can only lower the bound. Second, for non-Markov inputs the output is no longer Markov and the Venkataramanan-Tatikonda-Ramchandran analysis breaks down; we show that the residual length of the current input run turns the output into a hidden Markov chain, which extends the bound to every finite-support law. Third, its correction term counts only output runs formed from three input runs; we prove a larger correction that accounts for every output run formed by several input runs, which improves the published bound even for truncated geometric laws. Certified by interval arithmetic, the new bounds exceed all previously published deterministic lower bounds at every tabulated deletion probability, by up to $6.86\times10^{-3}$ bits per channel use and 6.8%. At large deletion probability the learned laws concentrate on run-length clusters with survivor counts about two standard deviations apart, like a pulse-amplitude constellation.
Comments22 pages, 6 figures, 4 tables, 3 algorithms. v2: rigorous chain-penalty correction for the VTR bound, verified interval-arithmetic evaluation of all bounds, complete proofs, and a reorganized presentation
