arXiv subjects

Jarosław Błasiok

Publications and source records attributed to Jarosław Błasiok.

24 records · Page 2Linked to original sources

arXiv2018

Polar Codes with exponentially small error at finite block length

We show that the entire class of polar codes (up to a natural necessary condition) converge to capacity at block lengths polynomial in the gap to capacity, while simultaneously achieving failure probabilities that are exponentially small in the block length (i.e., decoding fails with probability $\exp(-N^{Ω(1)})$ for codes of length $N$). Previously this combination was known only for one specific family within the class of polar codes, whereas we establish this whenever the polar code exhibits a condition necessary for any polarization. Our results adapt and strengthen a local analysis of polar codes due to the authors with Nakkiran and Rudra [Proc. STOC 2018]. Their analysis related the time-local behavior of a martingale to its global convergence, and this allowed them to prove that the broad class of polar codes converge to capacity at polynomial block lengths. Their analysis easily adapts to show exponentially small failure probabilities, provided the associated martingale, the ``Arikan martingale'', exhibits a corresponding strong local effect. The main contribution of this work is a much stronger local analysis of the Arikan martingale. This leads to the general result claimed above. In addition to our general result, we also show, for the first time, polar codes that achieve failure probability $\exp(-N^β)$ for any $β< 1$ while converging to capacity at block length polynomial in the gap to capacity. Finally we also show that the ``local'' approach can be combined with any analysis of failure probability of an arbitrary polar code to get essentially the same failure probability while achieving block length polynomial in the gap to capacity.

cs.IT↗

arXiv2018

The Generic Holdout: Preventing False-Discoveries in Adaptive Data Science

Adaptive data analysis has posed a challenge to science due to its ability to generate false hypotheses on moderately large data sets. In general, with non-adaptive data analyses (where queries to the data are generated without being influenced by answers to previous queries) a data set containing $n$ samples may support exponentially many queries in $n$. This number reduces to linearly many under naive adaptive data analysis, and even sophisticated remedies such as the Reusable Holdout (Dwork et. al 2015) only allow quadratically many queries in $n$. In this work, we propose a new framework for adaptive science which exponentially improves on this number of queries under a restricted yet scientifically relevant setting, where the goal of the scientist is to find a single (or a few) true hypotheses about the universe based on the samples. Such a setting may describe the search for predictive factors of some disease based on medical data, where the analyst may wish to try a number of predictive models until a satisfactory one is found. Our solution, the Generic Holdout, involves two simple ingredients: (1) a partitioning of the data into a exploration set and a holdout set and (2) a limited exposure strategy for the holdout set. An analyst is free to use the exploration set arbitrarily, but when testing hypotheses against the holdout set, the analyst only learns the answer to the question: "Is the given hypothesis true (empirically) on the holdout set?" -- and no more information, such as "how well" the hypothesis fits the holdout set. The resulting scheme is immediate to analyze, but despite its simplicity we do not believe our method is obvious, as evidenced by the many violations in practice. Our proposal can be seen as an alternative to pre-registration, and allows researchers to get the benefits of adaptive data analysis without the problems of adaptivity.

stat.ME↗

arXiv2018

Induced minors and well-quasi-ordering

A graph $H$ is an induced minor of a graph $G$ if it can be obtained from an induced subgraph of $G$ by contracting edges. Otherwise, $G$ is said to be $H$-induced minor-free. Robin Thomas showed that $K_4$-induced minor-free graphs are well-quasi-ordered by induced minors [Graphs without $K_4$ and well-quasi-ordering, Journal of Combinatorial Theory, Series B, 38(3):240 -- 247, 1985]. We provide a dichotomy theorem for $H$-induced minor-free graphs and show that the class of $H$-induced minor-free graphs is well-quasi-ordered by the induced minor relation if and only if $H$ is an induced minor of the gem (the path on 4 vertices plus a dominating vertex) or of the graph obtained by adding a vertex of degree 2 to the complete graph on 4 vertices. To this end we proved two decomposition theorems which are of independent interest. Similar dichotomy results were previously given for subgraphs by Guoli Ding in [Subgraphs and well-quasi-ordering, Journal of Graph Theory, 16(5):489--502, 1992] and for induced subgraphs by Peter Damaschke in [Induced subgraphs and well-quasi-ordering, Journal of Graph Theory, 14(4):427--435, 1990].

math.CO↗

arXiv2017

Continuous monitoring of $\ell_p$ norms in data streams

In insertion-only streaming, one sees a sequence of indices $a_1, a_2, \ldots, a_m\in [n]$. The stream defines a sequence of $m$ frequency vectors $x^{(1)},\ldots,x^{(m)}\in\mathbb{R}^n$ with $(x^{(t)})_i = |\{j : j\in[t], a_j = i\}|$. That is, $x^{(t)}$ is the frequency vector after seeing the first $t$ items in the stream. Much work in the streaming literature focuses on estimating some function $f(x^{(m)})$. Many applications though require obtaining estimates at time $t$ of $f(x^{(t)})$, for every $t\in[m]$. Naively this guarantee is obtained by devising an algorithm with failure probability $\ll 1/m$, then performing a union bound over all stream updates to guarantee that all $m$ estimates are simultaneously accurate with good probability. When $f(x)$ is some $\ell_p$ norm of $x$, recent works have shown that this union bound is wasteful and better space complexity is possible for the continuous monitoring problem, with the strongest known results being for $p=2$ [HTY14, BCIW16, BCINWW17]. In this work, we improve the state of the art for all $0<p<2$, which we obtain via a novel analysis of Indyk's $p$-stable sketch [Indyk06].

cs.DS↗

arXiv2016

ADAGIO: Fast Data-aware Near-Isometric Linear Embeddings

Many important applications, including signal reconstruction, parameter estimation, and signal processing in a compressed domain, rely on a low-dimensional representation of the dataset that preserves {\em all} pairwise distances between the data points and leverages the inherent geometric structure that is typically present. Recently Hedge, Sankaranarayanan, Yin and Baraniuk \cite{hedge2015} proposed the first data-aware near-isometric linear embedding which achieves the best of both worlds. However, their method NuMax does not scale to large-scale datasets. Our main contribution is a simple, data-aware, near-isometric linear dimensionality reduction method which significantly outperforms a state-of-the-art method \cite{hedge2015} with respect to scalability while achieving high quality near-isometries. Furthermore, our method comes with strong worst-case theoretical guarantees that allow us to guarantee the quality of the obtained near-isometry. We verify experimentally the efficiency of our method on numerous real-world datasets, where we find that our method ($<$10 secs) is more than 3\,000$\times$ faster than the state-of-the-art method \cite{hedge2015} ($>$9 hours) on medium scale datasets with 60\,000 data points in 784 dimensions. Finally, we use our method as a preprocessing step to increase the computational efficiency of a classification application and for speeding up approximate nearest neighbor queries.

stat.ML↗

arXiv2016

An improved analysis of the ER-SpUD dictionary learning algorithm

In "dictionary learning" we observe $Y = AX + E$ for some $Y\in\mathbb{R}^{n\times p}$, $A \in\mathbb{R}^{m\times n}$, and $X\in\mathbb{R}^{m\times p}$. The matrix $Y$ is observed, and $A, X, E$ are unknown. Here $E$ is "noise" of small norm, and $X$ is column-wise sparse. The matrix $A$ is referred to as a {\em dictionary}, and its columns as {\em atoms}. Then, given some small number $p$ of samples, i.e.\ columns of $Y$, the goal is to learn the dictionary $A$ up to small error, as well as $X$. The motivation is that in many applications data is expected to sparse when represented by atoms in the "right" dictionary $A$ (e.g.\ images in the Haar wavelet basis), and the goal is to learn $A$ from the data to then use it for other applications. Recently, [SWW12] proposed the dictionary learning algorithm ER-SpUD with provable guarantees when $E = 0$ and $m = n$. They showed if $X$ has independent entries with an expected $s$ non-zeroes per column for $1 \lesssim s \lesssim \sqrt{n}$, and with non-zero entries being subgaussian, then for $p\gtrsim n^2\log^2 n$ with high probability ER-SpUD outputs matrices $A', X'$ which equal $A, X$ up to permuting and scaling columns (resp.\ rows) of $A$ (resp.\ $X$). They conjectured $p\gtrsim n\log n$ suffices, which they showed was information theoretically necessary for {\em any} algorithm to succeed when $s \simeq 1$. Significant progress was later obtained in [LV15]. We show that for a slight variant of ER-SpUD, $p\gtrsim n\log(n/δ)$ samples suffice for successful recovery with probability $1-δ$. We also show that for the unmodified ER-SpUD, $p\gtrsim n^{1.99}$ samples are required even to learn $A, X$ with polynomially small success probability. This resolves the main conjecture of [SWW12], and contradicts the main result of [LV15], which claimed that $p\gtrsim n\log^4 n$ guarantees success whp.

cs.LG↗