On the clustering of rare codons and its effect on translation

The presence of clusters of rare codons is known to negatively impact the efficiency and accuracy of protein production. In this paper, we demonstrate a statistical method of identifying such clusters in the coding sequence of a gene. Using E. coli as our model organism, we show that genes having denser clusters tend to have lower protein yields

Wave packet frames generated by hyponormal operators on L2(R)L^2(\mathbb{R})

In this paper we study frame-like properties of a wave packet system by using hyponormal operators on $L^2(\mathbb{R})$. We present necessary and sufficient conditions in terms of relative hyponormality of operators for a system to be a wave packet frame in $L^2(\mathbb{R})$. A characterization of hyponormal operators by using tight wave packet frames is proved. This is different from a method proved by Djordjevi$\acute{c}$ by using the Moore-Penrose inverse of a bounded linear operator with a closed range. The linear combinations of wave packet frames generated by hyponormal operators are discussed

An approach to reachability analysis for feed-forward ReLU neural networks

We study the reachability problem for systems implemented as feed-forward neural networks whose activation function is implemented via ReLU functions. We draw a correspondence between establishing whether some arbitrary output can ever be outputed by a neural system and linear problems characterising a neural system of interest. We present a methodology to solve cases of practical interest by means of a state-of-the-art linear programs solver. We evaluate the technique presented by discussing the experimental results obtained by analysing reachability properties for a number of benchmarks in the literature

Vector-Valued (Super) Weaving Frames

Two frames $\{\phi_{i}\}_{i \in I}$ and $\{\psi_{i}\}_{i \in I}$ for a separable Hilbert space $H$ are woven if there are positive constants $A \leq B$ such that for every subset $\sigma \subset I$, the family $\{\phi_{i}\}_{i \in \sigma} \cup \{\psi_{i}\}_{i \in \sigma^{c}}$ is a frame for $H$ with frame bounds $A, B$. Bemrose et al. introduced weaving frames in separable Hilbert spaces and observed that weaving frames has potential applications in signal processing. Motivated by this, and the recent work of Balan in the direction of application of vector-valued frames (or superframes) in signal processing, we study vector-valued weaving frames. In this paper, first we give some fundamental properties of vector-valued weaving frames. It is shown that if a family of vector-valued frames is woven, then the corresponding family of frames for atomic spaces is woven, but the converse is not true. We present a technique for the construction of vector-valued woven frames from given woven frames for atomic spaces . Necessary and sufficient conditions for vector-valued weaving Riesz sequences are given. Several numerical examples are given to illustrate the results

Convergence rates for ordinal embedding

We prove optimal bounds for the convergence rate of ordinal embedding (also known as non-metric multidimensional scaling) in the 1-dimensional case. The examples witnessing optimality of our bounds arise from a result in additive number theory on sets of integers with no three-term arithmetic progressions. We also carry out some computational experiments aimed at developing a sense of what the convergence rate for ordinal embedding might look like in higher dimensions

Calculating the divided differences of the exponential function by addition and removal of inputs

We introduce a method for calculating the divided differences of the exponential function by means of addition and removal of items from the input list to the function. Our technique exploits a new identity related to divided differences recently derived by F. Zivcovich [Dolomites Research Notes on Approximation 12, 28-42 (2019)]. We show that upon adding an item to or removing an item from the input list of an already evaluated exponential, the re-evaluation of the divided differences can be done with only $O(s n)$ floating point operations and $O(s n)$ bytes of memory, where $[z_0,\dots,z_n]$ are the inputs and $s \propto \max_{i,j} |z_i - z_j|$. We demonstrate our algorithm's ability to deal with input lists that are orders-of-magnitude longer than the maximal capacities of the current state-of-the-art. We discuss in detail one practical application of our method: the efficient calculation of weights in the off-diagonal series expansion quantum Monte Carlo algorithm.Comment: 9 pages, 3 figure

Learning Low-Dimensional Metrics

This paper investigates the theoretical foundations of metric learning, focused on three key questions that are not fully addressed in prior work: 1) we consider learning general low-dimensional (low-rank) metrics as well as sparse metrics; 2) we develop upper and lower (minimax)bounds on the generalization error; 3) we quantify the sample complexity of metric learning in terms of the dimension of the feature space and the dimension/rank of the underlying metric;4) we also bound the accuracy of the learned metric relative to the underlying true generative metric. All the results involve novel mathematical approaches to the metric learning problem, and lso shed new light on the special case of ordinal embedding (aka non-metric multidimensional scaling).Comment: 19 pages, 3 figures, Accepted at NIPS 2017 - Edited version to match final submission to NIPS proceedings and correct several spelling error

On triangular numbers, forms of mixed type and their representation numbers

In \cite{ono}, K. Ono, S. Robins and P.T. Wahl considered the problem of determining formulas for the number of representations of a natural number $n$ by a sum of $k$ triangular numbers and derived many applications, including the one connecting these numbers with the number of representations of $n$ as a sum of $k$ odd square integers. They also obtained an application to the number of lattice points in the $k$-dimensional sphere. In this paper, we consider triangular numbers with positive integer coefficients. First we show that if the sum of these coefficients is a multiple of $8$, then the associated generating function gives rise to a modular form of integral weight (when even number of triangular numbers are taken). We then use the theory of modular forms to get the representation number formulas corresponding to the triangular numbers with coefficients. We also obtain several applications concerning the triangular numbers with coefficients similar to the ones obtained in \cite{ono}. In the second part of the paper, we consider more general mixed forms (as done in Xia-Ma-Tian \cite{xia}) and derive modular properties for the corresponding generating functions associated to these mixed forms. Using our method we deduce all the 21 formulas proved in \cite[Theorem 1.1]{xia} and show that our method of deriving the 21 formulas together with the $(p,k)$ parametrization of the generating functions of the three mixed forms imply the $(p,k)$ parametrization of the Eisenstein series $E_4(\tau)$ and its duplications. It is to be noted that the $(p,k)$ parametrization of $E_4$ and its duplications were derived by a different method by K. S. Williams and his co-authors. In the final section, we provide sample formulas for these representation numbers in the case of 4 and 6 variable forms.Comment: 36 pages, 17 tables, revised abstract, sections are reordered, some typos fixed,corollary 1.5 and remark 1.1 are combined, (p,k)-parametrization of Eisenstein series are correcte

Weaving K-frames in Hilbert Spaces

Gavruta introduced $K$-frames for Hilbert spaces to study atomic systems with respect to a bounded linear operator. There are many differences between K-frames and standard frames, so we study weaving properties of K-frames. Two frames $\{\phi_{i}\}_{i \in I}$ and $\{\psi_{i}\}_{i \in I}$ for a separable Hilbert space $\mathcal{H}$ are woven if there are positive constants $A \leq B$ such that for every subset $\sigma \subset I$, the family $\{\phi_{i}\}_{i \in \sigma} \cup \{\psi_{i}\}_{i \in \sigma^{c}}$ is a frame for $\mathcal{H}$ with frame bounds $A, B$. In this paper, we present necessary and sufficient conditions for weaving $K$-frames in Hilbert spaces. It is shown that woven $K$-frames and weakly woven $K$-frames are equivalent. Finally, sufficient conditions for Paley-Wiener type perturbation of weaving $K$-frames are given

A Bandit Approach to Multiple Testing with False Discovery Control

We propose an adaptive sampling approach for multiple testing which aims to maximize statistical power while ensuring anytime false discovery control. We consider $n$ distributions whose means are partitioned by whether they are below or equal to a baseline (nulls), versus above the baseline (actual positives). In addition, each distribution can be sequentially and repeatedly sampled. Inspired by the multi-armed bandit literature, we provide an algorithm that takes as few samples as possible to exceed a target true positive proportion (i.e. proportion of actual positives discovered) while giving anytime control of the false discovery proportion (nulls predicted as actual positives). Our sample complexity results match known information theoretic lower bounds and through simulations we show a substantial performance improvement over uniform sampling and an adaptive elimination style algorithm. Given the simplicity of the approach, and its sample efficiency, the method has promise for wide adoption in the biological sciences, clinical testing for drug discovery, and online A/B/n testing problems