About Adaptive Coding on Countable Alphabets: Max-Stable Envelope Classes

In this paper, we study the problem of lossless universal source coding for stationary memoryless sources on countably infinite alphabets. This task is generally not achievable without restricting the class of sources over which universality is desired. Building on our prior work, we propose natural families of sources characterized by a common dominating envelope. We particularly emphasize the notion of adaptivity, which is the ability to perform as well as an oracle knowing the envelope, without actually knowing it. This is closely related to the notion of hierarchical universal source coding, but with the important difference that families of envelope classes are not discretely indexed and not necessarily nested. Our contribution is to extend the classes of envelopes over which adaptive universal source coding is possible, namely by including max-stable (heavy-tailed) envelopes which are excellent models in many applications, such as natural language modeling. We derive a minimax lower bound on the redundancy of any code on such envelope classes, including an oracle that knows the envelope. We then propose a constructive code that does not use knowledge of the envelope. The code is computationally efficient and is structured to use an {E}xpanding {T}hreshold for {A}uto-{C}ensoring, and we therefore dub it the \textsc{ETAC}-code. We prove that the \textsc{ETAC}-code achieves the lower bound on the minimax redundancy within a factor logarithmic in the sequence length, and can be therefore qualified as a near-adaptive code over families of heavy-tailed envelopes. For finite and light-tailed envelopes the penalty is even less, and the same code follows closely previous results that explicitly made the light-tailed assumption. Our technical results are founded on methods from regular variation theory and concentration of measure

Large alphabets: Finite, infinite, and scaling models

How can we effectively model situations with large alphabets? On a pragmatic level, any engineered system, be it for inference, communication, or encryption, requires working with a finite number of symbols. Therefore, the most straight-forward model is a finite alphabet. However, to emphasize the disproportionate size of the alphabet, one may want to compare its finite size with the length of data at hand. More generally, this gives rise to scaling models that strive to capture regimes of operation where one anticipates such imbalance. Large alphabets may also be idealized as infinite. The caveat then is that such generality strips away many of the convenient machinery of finite settings. However, some of it may be salvaged by refocusing the tasks of interest, such as by moving from sequence to pattern compression, or by minimally restricting the classes of infinite models, such as via tail properties. In this paper we present an overview of models for large alphabets, some recent results, and possible directions in this area

Rare Probability Estimation under Regularly Varying Heavy Tails

This paper studies the problem of estimating the probability of symbols that have occurred very rarely, in samples drawn independently from an unknown, possibly infinite, discrete distribution. In particular, we study the multiplicative consistency of estimators, defined as the ratio of the estimate to the true quantity converging to one. We first show that the classical Good-Turing estimator is not universally consistent in this sense, despite enjoying favorable additive properties. We then use Karamata's theory of regular variation to prove that regularly varying heavy tails are sufficient for consistency. At the core of this result is a multiplicative concentration that we establish both by extending the McAllester-Ortiz additive concentration for the missing mass to all rare probabilities and by exploiting regular variation. We also derive a family of estimators which, in addition to being consistent, address some of the shortcomings of the Good-Turing estimator. For example, they perform smoothing implicitly and have the absolute discounting structure of many heuristic algorithms. This also establishes a discrete parallel to extreme value theory, and many of the techniques therein can be adapted to the framework that we set forth.National Science Foundation (U.S.) (Grant 6922470)United States. Office of Naval Research (Grant 6918937

Central spindle self-organization and cytokinesis in artificially activated sea urchin eggs

Author Posting. © Marine Biological Laboratory, 2016. This article is posted here by permission of Marine Biological Laboratory for personal use, not for redistribution. The definitive version was published in Biological Bulletin 230, no.2 (2016): 85-95.The ability of microtubules of the mitotic apparatus to control the positioning and initiation of the cleavage furrow during cytokinesis was first established from studies on early echinoderm embryos. However, the identity of the microtubule population that imparts cytokinetic signaling is unclear. The two main––and not necessarily mutually exclusive–– candidates are the central spindle and the astral rays. In the present study, we examined cytokinesis in ammonia-activated sea urchin eggs, which lack paternally derived centrosomes and undergo mitosis mediated by unusual anastral, bipolar mini-spindles. Live cell imaging and immunolabeling for microtubules and the centralspindlin constituent and kinesin-related protein, MKLP1, demonstrated that furrowing in ammonia-activated eggs was associated with aligned arrays of centralspindlin-linked, opposed bundles of antiparallel microtubules. These autonomous, zipper- like arrays were not associated with a mitotic apparatus, but did possess characteristics similar to the central spindle region of control, fertilized embryos. Our results highlight the self-organizing nature of the central spindle region and its ability to induce cytokinesis-like furrowing, even in the absence of a complete mitotic apparatus.This research was supported by student/faculty summer research grants from the Dickinson College Research and Development Committee to JHH; Laura and Arthur Colwin Summer Research Fellowships from the MBL to JHH and CBS; a National Science Foundation Major Research Instrumentation grant to JHH (MRI-0320606); and a NSF collaborative research grant to JHH (MCB-1412688) and to CBS (MCB- 1412734)

The participation of the Armenian community in ottoman public life in Eastern Anatolia and Syria, 1860-1908

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Telegram from Mesrob Ashjian, Reverend Archbishop of the Armenian Apostolic Church of America, to Geraldine Ferraro

Congratulatory telegram from Mesrob Ashjian, Reverend Archbishop of the Armenian Apostolic Church of America, to Geraldine Ferraro. Archbishop Ashjian calls Ferraro a sincere friend of the Armenian people. Includes standard response letter.https://ir.lawnet.fordham.edu/vice_presidential_campaign_correspondence_1984_new_york/1267/thumbnail.jp