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The ability of large language models (LLMs) to execute complex instructions is essential for their real-world applications. However, several recent studies indicate that LLMs struggle with challenging instructions. In this paper, we propose Evolutionary Contrastive Distillation (ECD), a novel method for generating high-quality synthetic preference data designed to enhance the complex instruction-following capability of language models. ECD generates data that specifically illustrates the difference between a response that successfully follows a set of complex instructions and a response that is high-quality, but nevertheless makes some subtle mistakes. This is done by prompting LLMs to progressively evolve simple instructions to more complex instructions. When the complexity of an instruction is increased, the original successful response to the original instruction becomes a "hard negative" response for the new instruction, mostly meeting requirements of the new instruction, but barely missing one or two. By pairing a good response with such a hard negative response, and employing contrastive learning algorithms such as DPO, we improve language models' ability to follow complex instructions. Empirically, we observe that our method yields a 7B model that exceeds the complex instruction-following performance of current SOTA 7B models and is competitive even with open-source 70B models.

In the late stages of accretion leading up to the formation of planetesimals, particles grew to pebbles the size of 1-mm to tens of cm. That is the same size range that dominates the present-day comet mass loss. Meteoroids that size cause visible meteors on Earth. Here, we hypothesize that the size distribution and the physical and chemical properties of young meteoroid streams still contain information about the conditions in the solar nebula during these late stages of accretion. From observations of 47 young meteor showers, we find that freshly ejected meteoroids from long-period comets tend to have low bulk density and are distributed with equal surface area per log-mass interval (magnitude distribution index chi ~ 1.85), suggesting gentle accretion conditions. Jupiter-family comets, on the other hand, mostly produce meteoroids twice as dense and distributed with a steeper chi ~ 2.15 or even chi ~ 2.5, which implies that those pebbles grew from particles fragmenting in a collisional cascade or by catastrophic collisions, respectively. Both comet populations contain an admixture of compact materials that are sometimes sodium-poor, but Jupiter-family comets show a higher percentage (~8% on average) than long-period comet showers (~4%), and a wider range. While there are exceptions in both groups, the implication is that most long-period comets formed under gentle particle growth conditions, possibly near the 30 AU edge of the Trans Neptunian Disk, while most Jupiter family comets formed closer to the Sun where pebbles reached or passed the fragmentation barrier. This is possible if the Scattered Disk represents all objects scattered by Neptune during its migration, while the present-day outer Oort cloud formed only during and after the Sun had moved away from sibling stars., Comment: 82 pages, 14 figures, 3 tables

The latest result from The ORGAN Experiment, an axion haloscope is presented. This iteration of the experiment operated at millikelvin temperatures using a flux-driven Josephson parametric amplifier (JPA) for reduced noise, along with various other upgrades over previous iterations. Covering the $25.45 - 26.27\,\mu\text{eV}$ ($6.15-6.35$ GHz) mass (frequency) range, this near-quantum limited phase of ORGAN employs a conducting rod resonator and a 7-T solenoidal magnet to place the most sensitive exclusion limits on axion-photon coupling in the range to date, with $|g_{a\gamma\gamma}| \gtrsim 2.8\times10^{-13}$ at a 95% confidence level., Comment: 9 pages, 5 figures

Recent work of the author established dual representation theorems for certain vector spaces that arise in an important article of Allcock and Vaaler. These results constructed an object called a consistent map which acts like a measure on the set of places of $\bar{\mathbb Q}$, but is not a Borel measure on this space. We describe the appropriate ring of sets $\mathcal R$ for which every consistent map arises from a measure on $\mathcal R$. We further obtain the conditions under which a consistent map may be extended to a measure on the smallest algebra containing $\mathcal R$.

The ability of Augmented Reality to overcome the bias of single stories through multidimensionality is explored in the artifacts of a youth gun violence prevention project and its goal of narrative change., Comment: Presented at CHI 2024 (arXiv:2404.05889)

Out-of-distribution (OOD) generalization is critical for machine learning models deployed in the real world. However, achieving this can be fundamentally challenging, as it requires the ability to learn invariant features across different domains or environments. In this paper, we propose a novel framework HYPO (HYPerspherical OOD generalization) that provably learns domain-invariant representations in a hyperspherical space. In particular, our hyperspherical learning algorithm is guided by intra-class variation and inter-class separation principles -- ensuring that features from the same class (across different training domains) are closely aligned with their class prototypes, while different class prototypes are maximally separated. We further provide theoretical justifications on how our prototypical learning objective improves the OOD generalization bound. Through extensive experiments on challenging OOD benchmarks, we demonstrate that our approach outperforms competitive baselines and achieves superior performance. Code is available at https://github.com/deeplearning-wisc/hypo., Comment: The conference version of this paper is published at ICLR 2024; First two authors contributed equally

This brief covers the Advancing Early Education Collaborative (AEEC)'s efforts to address the lack of equitable opportunities for early childhood educators and the quality of early child care and childhood education in DC. The AEEC has the goals of enhancing early education career pathways for Black and Latina women in Wards 7 and 8 and improving the overall quality of early education in Washington, DC. Beginning with an overview of the project and its early accomplishments, this brief highlights findings from a focus group with program participants from American University and concludes with reflections on the AEEC's successes and challenges thus far, including opportunities to incorporate lessons learned into the collaborative's activities in year two.

A 2009 article of Allcock and Vaaler examined the vector space $\mathcal G := \overline{\mathbb Q}^\times/\overline{\mathbb Q}^\times_{\mathrm{tors}}$ over $\mathbb Q$, describing its completion with respect to the Weil height as a certain $L^1$ space. By involving an object called a consistent map, the author began efforts to establish representation theorems for the duals of spaces related to $\mathcal G$. Specifically, we provided representation theorems for the algebraic and continuous duals of $\overline{\mathbb Q}^\times/{\overline{\mathbb Z}}^\times$. We explore further applications of consistent maps to provide representation theorems for duals of other spaces arising in the work of Allcock and Vaaler. In particular, we establish a connection between consistent maps and locally constant functions on the space of places of $\overline{\mathbb Q}$. We further apply our new results to recover, as a corollary, a main theorem of our previous work.

Metallic pipes and other containers are widely used to store and transport toxic gases and liquids. Various sensors have been designed to monitor the environment inside metallic pipes and containers, such as pressure, liquid-level, and chemical sensors. Moreover, sensors are also used to inspect and detect pipe leakages. However, sensors are usually placed outside of metallic pipes and containers and use ultrasound to monitor the internal unseen environment. This is mainly due to the fact that internal sensors cannot communicate with external data sinks without cables, but using cables can dramatically affect the metal-sealed structure. Wireless communication is desirable to communicate with internal sensors, but it experiences high attenuation losses since metal can block wireless signals due to its high conductivity. This paper develops a low-cost through-metal communication system prototype using off-the-shelf electronic components. The system is fully reconfigurable, and arbitrary modulation and coding schemes can be implemented. We design the transmit module which includes a signal processing microcontroller, an amplifier, and a transmit coil, and the receive module which includes a receive coil, an amplifier, and a microcontroller with demodulation algorithms and bit-error-rate (BER) calculations. The performance of the prototype is evaluated using various symbol rates, distances, and transmission power. The results show that the communication system can achieve a 500 bps data rate with 0.01 BER and 3.4 cm communication range when penetrating an Aluminum pipe with 7 mm thickness.

A recent observational study found that the projected spatial distributions of the satellites of bright, isolated host galaxies tend to be lopsided with respect to the locations of the hosts. Here, we examine the spatial distributions of the satellites of a large number of bright, isolated host galaxies that were obtained from mock redshift surveys of a LCDM simulation. Host galaxies and their satellites were identified using selection criteria that are identical to those used in the observational study, allowing a direct comparison of the results for the simulated and observed systems. To characterize the spatial distribution of the satellites, we adopt two statistics: [1] the pairwise clustering of the satellites and [2] the Mean Resultant Length. In agreement with the observational study, we find a strong tendency for satellites in the simulation to be located on the same side of their host, and the signal is most pronounced for the satellites of blue hosts. These lopsided satellite distributions are not solely attributable to incompleteness of the observed satellite catalog or the presence of objects that have been falsely identified as satellites. In addition, satellites that joined their hosts' halos in the distant past (> 8 Gyr) show a pronounced lopsidedness in their spatial distributions and, therefore, the lopsidedness is not solely attributable to late-time accretion of satellites.

SBcoyote is an open-source cross-platform biochemical reaction viewer and editor released under the liberal MIT license. It is written in Python and uses wxPython to implement the GUI and the drawing canvas. It supports the visualization and editing of compartments, species, and reactions. It includes many options to stylize each of these components. For instance, species can be in different colors and shapes. Other core features include the ability to create alias nodes, alignment of groups of nodes, network zooming, as well as an interactive bird-eye view of the network to allow easy navigation on large networks. A unique feature of the tool is the extensive Python plugin API, where third-party developers can include new functionality. To assist third-party plugin developers, we provide a variety of sample plugins, including, random network generation, a simple auto layout tool, export to Antimony, export SBML, import SBML, etc. Of particular interest are the export and import SBML plugins since these support the SBML level 3 layout and render standard, which is exchangeable with other software packages. Plugins are stored in a GitHub repository, and an included plugin manager can retrieve and install new plugins from the repository on demand. Plugins have version metadata associated with them to make it install plugin updates. Availability: https://github.com/sys-bio/SBcoyote.

Out-of-distribution (OOD) detection is important for machine learning models deployed in the wild. Recent methods use auxiliary outlier data to regularize the model for improved OOD detection. However, these approaches make a strong distributional assumption that the auxiliary outlier data is completely separable from the in-distribution (ID) data. In this paper, we propose a novel framework that leverages wild mixture data, which naturally consists of both ID and OOD samples. Such wild data is abundant and arises freely upon deploying a machine learning classifier in their natural habitats. Our key idea is to formulate a constrained optimization problem and to show how to tractably solve it. Our learning objective maximizes the OOD detection rate, subject to constraints on the classification error of ID data and on the OOD error rate of ID examples. We extensively evaluate our approach on common OOD detection tasks and demonstrate superior performance., Comment: Accepted to International Conference on Machine Learning (ICML) 2022

Active learning is a label-efficient approach to train highly effective models while interactively selecting only small subsets of unlabelled data for labelling and training. In "open world" settings, the classes of interest can make up a small fraction of the overall dataset -- most of the data may be viewed as an out-of-distribution or irrelevant class. This leads to extreme class-imbalance, and our theory and methods focus on this core issue. We propose a new strategy for active learning called GALAXY (Graph-based Active Learning At the eXtrEme), which blends ideas from graph-based active learning and deep learning. GALAXY automatically and adaptively selects more class-balanced examples for labeling than most other methods for active learning. Our theory shows that GALAXY performs a refined form of uncertainty sampling that gathers a much more class-balanced dataset than vanilla uncertainty sampling. Experimentally, we demonstrate GALAXY's superiority over existing state-of-art deep active learning algorithms in unbalanced vision classification settings generated from popular datasets.

We consider interactive learning in the realizable setting and develop a general framework to handle problems ranging from best arm identification to active classification. We begin our investigation with the observation that agnostic algorithms \emph{cannot} be minimax-optimal in the realizable setting. Hence, we design novel computationally efficient algorithms for the realizable setting that match the minimax lower bound up to logarithmic factors and are general-purpose, accommodating a wide variety of function classes including kernel methods, H{\"o}lder smooth functions, and convex functions. The sample complexities of our algorithms can be quantified in terms of well-known quantities like the extended teaching dimension and haystack dimension. However, unlike algorithms based directly on those combinatorial quantities, our algorithms are computationally efficient. To achieve computational efficiency, our algorithms sample from the version space using Monte Carlo "hit-and-run" algorithms instead of maintaining the version space explicitly. Our approach has two key strengths. First, it is simple, consisting of two unifying, greedy algorithms. Second, our algorithms have the capability to seamlessly leverage prior knowledge that is often available and useful in practice. In addition to our new theoretical results, we demonstrate empirically that our algorithms are competitive with Gaussian process UCB methods.

Let $\overline{\mathbb Q}$ be an algebraic closure of $\mathbb Q$ and let $A$ denote the ring of algebraic integers in $\overline{\mathbb Q}$. If $\mathcal S = \overline{\mathbb Q}^\times/A^\times$ then $\mathcal S$ is a vector space over $\mathbb Q$. We provide a complete classification all elements in the algebraic dual $\mathcal S^*$ of $\mathcal S$ in terms of another $\mathbb Q$-vector space called the space of consistent maps. With an appropriate norm on $\mathcal S$, we further classify the continuous elements of $\mathcal S^*$. As applications of our results, we classify extensions of the prime Omega function to $\mathcal S$ and discuss a natural action of the absolute Galois group $\mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q)$ on $\mathcal S$., Comment: 21 pages

For each algebraic number $\alpha$ and each positive real number $t$, the $t$-metric Mahler measure $m_t(\alpha)$ creates an extremal problem whose solution varies depending on the value of $t$. The second author studied the points $t$ at which the solution changes, called {\it exceptional points for $\alpha$}. Although each algebraic number has only finitely many exceptional points, it is conjectured that, for every $N \in \mathbb N$, there exists a number having at least $N$ exceptional points. In this article, we describe a polynomial time algorithm for establishing the existence of numbers with at least $N$ exceptional points. Our work constitutes an improvement over the best known existing algorithm which requires exponential time. We apply our main result to show that there exist numbers with at least $37$ exceptional points, another improvement over previous work which was only able to reach $11$ exceptional points., Comment: 12 pages

We report the discovery of pulsations in the extremely low mass (ELM), likely helium-core white dwarf GD 278 via ground- and space-based photometry. GD 278 was observed by the Transiting Exoplanet Survey Satellite (TESS) in Sector 18 at a 2-min cadence for roughly 24 d. The TESS data reveal at least 19 significant periodicities between 2447-6729 s, one of which is the longest pulsation period ever detected in a white dwarf. Previous spectroscopy found that this white dwarf is in a 4.61 hr orbit with an unseen >0.4 solar-mass companion and has Teff = 9230 +/- 100 K and log(g) = 6.627 +/- 0.056, which corresponds to a mass of 0.191 +/- 0.013 solar mass. Patterns in the TESS pulsation frequencies from rotational splittings appear to reveal a stellar rotation period of roughly 10 hr, making GD 278 the first ELM white dwarf with a measured rotation rate. The patterns inform our mode identification for asteroseismic fits, which unfortunately do not reveal a global best-fit solution. Asteroseismology reveals two main solutions roughly consistent with the spectroscopic parameters of this ELM white dwarf, but with vastly different hydrogen-layer masses; future seismic fits could be further improved by using the stellar parallax. GD 278 is now the tenth known pulsating ELM white dwarf; it is only the fifth known to be in a short-period binary, but is the first with extended, space-based photometry., Comment: 14 pages, 10 figures, accepted for publication in the Astrophysical Journal

We introduce the model selection problem in pure exploration linear bandits, where the learner needs to adapt to the instance-dependent complexity measure of the smallest hypothesis class containing the true model. We design algorithms in both fixed confidence and fixed budget settings with near instance optimal guarantees. The core of our algorithms is a new optimization problem based on experimental design that leverages the geometry of the action set to identify a near-optimal hypothesis class. Our fixed budget algorithm is developed based on a novel selection-validation procedure, which provides a new way to study the understudied fixed budget setting (even without the added challenge of model selection). We adapt our algorithms, in both fixed confidence and fixed budget settings, to problems with model misspecification.

We consider active learning for binary classification in the agnostic pool-based setting. The vast majority of works in active learning in the agnostic setting are inspired by the CAL algorithm where each query is uniformly sampled from the disagreement region of the current version space. The sample complexity of such algorithms is described by a quantity known as the disagreement coefficient which captures both the geometry of the hypothesis space as well as the underlying probability space. To date, the disagreement coefficient has been justified by minimax lower bounds only, leaving the door open for superior instance dependent sample complexities. In this work we propose an algorithm that, in contrast to uniform sampling over the disagreement region, solves an experimental design problem to determine a distribution over examples from which to request labels. We show that the new approach achieves sample complexity bounds that are never worse than the best disagreement coefficient-based bounds, but in specific cases can be dramatically smaller. From a practical perspective, the proposed algorithm requires no hyperparameters to tune (e.g., to control the aggressiveness of sampling), and is computationally efficient by means of assuming access to an empirical risk minimization oracle (without any constraints). Empirically, we demonstrate that our algorithm is superior to state of the art agnostic active learning algorithms on image classification datasets.

In recent years methods from optimal linear experimental design have been leveraged to obtain state of the art results for linear bandits. A design returned from an objective such as $G$-optimal design is actually a probability distribution over a pool of potential measurement vectors. Consequently, one nuisance of the approach is the task of converting this continuous probability distribution into a discrete assignment of $N$ measurements. While sophisticated rounding techniques have been proposed, in $d$ dimensions they require $N$ to be at least $d$, $d \log(\log(d))$, or $d^2$ based on the sub-optimality of the solution. In this paper we are interested in settings where $N$ may be much less than $d$, such as in experimental design in an RKHS where $d$ may be effectively infinite. In this work, we propose a rounding procedure that frees $N$ of any dependence on the dimension $d$, while achieving nearly the same performance guarantees of existing rounding procedures. We evaluate the procedure against a baseline that projects the problem to a lower dimensional space and performs rounding which requires $N$ to just be at least a notion of the effective dimension. We also leverage our new approach in a new algorithm for kernelized bandits to obtain state of the art results for regret minimization and pure exploration. An advantage of our approach over existing UCB-like approaches is that our kernel bandit algorithms are also robust to model misspecification.

In this paper we propose a novel experimental design-based algorithm to minimize regret in online stochastic linear and combinatorial bandits. While existing literature tends to focus on optimism-based algorithms--which have been shown to be suboptimal in many cases--our approach carefully plans which action to take by balancing the tradeoff between information gain and reward, overcoming the failures of optimism. In addition, we leverage tools from the theory of suprema of empirical processes to obtain regret guarantees that scale with the Gaussian width of the action set, avoiding wasteful union bounds. We provide state-of-the-art finite time regret guarantees and show that our algorithm can be applied in both the bandit and semi-bandit feedback regime. In the combinatorial semi-bandit setting, we show that our algorithm is computationally efficient and relies only on calls to a linear maximization oracle. In addition, we show that with slight modification our algorithm can be used for pure exploration, obtaining state-of-the-art pure exploration guarantees in the semi-bandit setting. Finally, we provide, to the best of our knowledge, the first example where optimism fails in the semi-bandit regime, and show that in this setting our algorithm succeeds.

We investigate the spatial distribution of the satellites of bright, isolated host galaxies. In agreement with previous studies, we find that, on average, the satellites of red hosts are found preferentially close to their hosts' major axes, while the satellites of blue hosts are distributed isotropically. We compute the pairwise clustering of the satellites and find a strong tendency for pairs of satellites to be located on the same side of their host, resulting in lopsided distributions. The signal is most pronounced for the satellites of blue hosts, where the number of pairs on the same side of their host exceeds the number of pairs on opposite sides of their by a factor of 1.8 +/- 0.1. For the satellites of red hosts, the number of pairs on the same side of their host exceeds the number of pairs on opposite sides of their host by a factor of 1.08 +/- 0.03. Satellites that are far from their hosts (r_p > 300 kpc) show a strong preference for being located on the same side of their hosts; satellites that are near to their hosts (r_p < 100 kpc) show a weak preference for being located on opposite sides of their hosts. While lopsided distributions have been found previously for the satellites of bright pairs of galaxies, ours is the first study to find lopsided distributions for the satellites of isolated bright galaxies., Comment: 9 pages, 5 figures, accepted for publication in ApJ Letters

Sentiment analysis is one of the fastest spreading research areas in computer science, making it challenging to keep track of all the activities in the area. We present a customer feedback reviews on product, where we utilize opinion mining, text mining and sentiments, which has affected the surrounded world by changing their opinion on a specific product. Data used in this study are online product reviews collected from Amazon.com. We performed a comparative sentiment analysis of retrieved reviews. This research paper provides you with sentimental analysis of various smart phone opinions on smart phones dividing them Positive, Negative and Neutral Behaviour.

Modern technological era has reshaped traditional lifestyle in several domains. The medium of publishing news and events has become faster with the advancement of Information Technology. IT has also been flooded with immense amounts of data, which is being published every minute of every day, by millions of users, in the shape of comments, blogs, news sharing through blogs, social media micro-blogging websites and many more. Manual traversal of such huge data is a challenging job, thus, sophisticated methods are acquired to perform this task automatically and efficiently. News reports events that comprise of emotions - good, bad, neutral. Sentiment analysis is utilized to investigate human emotions present in textual information. This paper presents a lexicon-based approach for sentiment analysis of news articles. The experiments have been performed on BBC news data set, which expresses the applicability and validation of the adopted approach.

Nowadays, people from all around the world use social media sites to share information. Twitter for example is a platform in which users send, read posts known as tweets and interact with different communities. Users share their daily lives, post their opinions on everything such as brands and places. Companies can benefit from this massive platform by collecting data related to opinions on them. The aim of this paper is to present a model that can perform sentiment analysis of real data collected from Twitter. Data in Twitter is highly unstructured which makes it difficult to analyze. However, our proposed model is different from prior work in this field because it combined the use of supervised and unsupervised machine learning algorithms. The process of performing sentiment analysis as follows: Tweet extracted directly from Twitter API, then cleaning and discovery of data performed. After that the data were fed into several models for the purpose of training. Each tweet extracted classified based on its sentiment whether it is a positive, negative or neutral. Data were collected on two subjects McDonalds and KFC to show which restaurant has more popularity. Different machine learning algorithms were used. The result from these models were tested using various testing metrics like cross validation and f-score. Moreover, our model demonstrates strong performance on mining texts extracted directly from Twitter.

Many active learning and search approaches are intractable for large-scale industrial settings with billions of unlabeled examples. Existing approaches search globally for the optimal examples to label, scaling linearly or even quadratically with the unlabeled data. In this paper, we improve the computational efficiency of active learning and search methods by restricting the candidate pool for labeling to the nearest neighbors of the currently labeled set instead of scanning over all of the unlabeled data. We evaluate several selection strategies in this setting on three large-scale computer vision datasets: ImageNet, OpenImages, and a de-identified and aggregated dataset of 10 billion images provided by a large internet company. Our approach achieved similar mean average precision and recall as the traditional global approach while reducing the computational cost of selection by up to three orders of magnitude, thus enabling web-scale active learning.

This paper proposes near-optimal algorithms for the pure-exploration linear bandit problem in the fixed confidence and fixed budget settings. Leveraging ideas from the theory of suprema of empirical processes, we provide an algorithm whose sample complexity scales with the geometry of the instance and avoids an explicit union bound over the number of arms. Unlike previous approaches which sample based on minimizing a worst-case variance (e.g. G-optimal design), we define an experimental design objective based on the Gaussian-width of the underlying arm set. We provide a novel lower bound in terms of this objective that highlights its fundamental role in the sample complexity. The sample complexity of our fixed confidence algorithm matches this lower bound, and in addition is computationally efficient for combinatorial classes, e.g. shortest-path, matchings and matroids, where the arm sets can be exponentially large in the dimension. Finally, we propose the first algorithm for linear bandits in the the fixed budget setting. Its guarantee matches our lower bound up to logarithmic factors.

We consider two multi-armed bandit problems with $n$ arms: (i) given an $\epsilon > 0$, identify an arm with mean that is within $\epsilon$ of the largest mean and (ii) given a threshold $\mu_0$ and integer $k$, identify $k$ arms with means larger than $\mu_0$. Existing lower bounds and algorithms for the PAC framework suggest that both of these problems require $\Omega(n)$ samples. However, we argue that these definitions not only conflict with how these algorithms are used in practice, but also that these results disagree with intuition that says (i) requires only $\Theta(\frac{n}{m})$ samples where $m = |\{ i : \mu_i > \max_{i \in [n]} \mu_i - \epsilon\}|$ and (ii) requires $\Theta(\frac{n}{m}k)$ samples where $m = |\{ i : \mu_i > \mu_0 \}|$. We provide definitions that formalize these intuitions, obtain lower bounds that match the above sample complexities, and develop explicit, practical algorithms that achieve nearly matching upper bounds.

Suppose that $p$ is an odd prime and $\genfrac{(}{)}{}{}{\cdot}{p}$ denotes the Legendre symbol modulo $p$. If $p$ is has the form $p= n^2+1$ then one easily verifies that $\genfrac{(}{)}{}{}{a}{p} = \genfrac{(}{)}{}{}{-a}{p}$ for all $a\in \mathbb Z/p\mathbb Z$. We identify various symmetry properties of sequential matrices over $\mathbb Z/(n^2+1)\mathbb Z$ regardless of whether $n^2+1$ is prime. We deduce from these results a collection of symmetries involving Jacobi symbol modulo $n^2+1$ which generalize our above observation on the Legendre symbol.

The ad\`ele ring $\mathbb A_K$ of a global field $K$ is a locally compact, metrizable topological ring which is complete with respect to any invariant metric on $\mathbb A_K$. For a fixed global field $F$ and a possibly infinite algebraic extension $E/F$, there is a natural partial ordering on $\{\mathbb A_K:F\subseteq K\subseteq E\}$. Therefore, we may form the direct limit \[ \mathbb A_E = \varinjlim \mathbb A_K \] which provides one possible generalization of ad\`ele rings to arbitrary algebraic extensions $E/F$. In the case where $E/F$ is Galois, we define an alternate generalization of the ad\`eles, denoted $\bar{\mathbb V}_E$, to be a certain metrizable topological ring of continuous functions on the set of places of $E$. We show that $\bar{\mathbb V}_E$ is isomorphic to the completion of $\mathbb A_E$ with respect to any invariant metric and use this isomorphism to establish several topological properties of $\mathbb A_E$., Comment: 18 pages

Many machine learning problems can be characterized by mutual contamination models. In these problems, one observes several random samples from different convex combinations of a set of unknown base distributions and the goal is to infer these base distributions. This paper considers the general setting where the base distributions are defined on arbitrary probability spaces. We examine three popular machine learning problems that arise in this general setting: multiclass classification with label noise, demixing of mixed membership models, and classification with partial labels. In each case, we give sufficient conditions for identifiability and present algorithms for the infinite and finite sample settings, with associated performance guarantees., Comment: Published in JMLR. Subsumes arXiv:1602.06235

We consider the task of collaborative preference completion: given a pool of items, a pool of users and a partially observed item-user rating matrix, the goal is to recover the \emph{personalized ranking} of each user over all of the items. Our approach is nonparametric: we assume that each item $i$ and each user $u$ have unobserved features $x_i$ and $y_u$, and that the associated rating is given by $g_u(f(x_i,y_u))$ where $f$ is Lipschitz and $g_u$ is a monotonic transformation that depends on the user. We propose a $k$-nearest neighbors-like algorithm and prove that it is consistent. To the best of our knowledge, this is the first consistency result for the collaborative preference completion problem in a nonparametric setting. Finally, we demonstrate the performance of our algorithm with experiments on the Netflix and Movielens datasets., Comment: AISTATS 2018

For an algebraic number $\alpha$, the metric Mahler measure $m_1(\alpha)$ was first studied by Dubickas and Smyth in 2001 and was later generalized to the $t$-metric Mahler measure $m_t(\alpha)$ by the author in 2010. The definition of $m_t(\alpha)$ involves taking an infimum over a certain collection $N$-tuples of points in $\overline{\mathbb Q}$, and from previous work of Jankauskas and the author, the infimum in the definition of $m_t(\alpha)$ is attained by rational points when $\alpha\in \mathbb Q$. As a consequence of our main theorem in this article, we obtain an analog of this result when $\mathbb Q$ is replaced with any imaginary quadratic number field of class number equal to $1$. Further, we study examples of other number fields to which our methods may be applied, and we establish various partial results in those cases., Comment: 12 pages

We report on the characterization of silica aerogel thermal optical nonlinearity, obtained by z-scan technique. The results show that typical silica aerogels have nonlinear optical coefficient similar to that of glass $(\simeq 10^{-12} $m$^2/$W), with negligible optical nonlinear absorption. The non\-li\-near coefficient can be increased to values in the range of $10^{-10} $m$^2/$W by embedding an absorbing dye in the aerogel. This value is one order of magnitude higher than that observed in the pure dye and in typical highly nonlinear materials like liquid crystals., Comment: 5 pages, 5 figures, Published by AIP Publishing

If $\alpha$ is a non-zero algebraic number, we let $m(\alpha)$ denote the Mahler measure of the minimal polynomial of $\alpha$ over $\mathbb Z$. A series of articles by Dubickas and Smyth, and later by the author, develop a modified version of the Mahler measure called the $t$-metric Mahler measure, denoted $m_t(\alpha)$. For fixed $\alpha\in \bar{\mathbb Q}$, the map $t\mapsto m_t(\alpha)$ is continuous, and moreover, is infinitely differentiable at all but finitely many points, called {\it exceptional points} for $\alpha$. It remains open to determine whether there is a sequence of elements $\alpha_n\in \bar{\mathbb Q}$ such that the number of exceptional points for $\alpha_n$ tends to $\infty$ as $n\to \infty$. We utilize a connection with the Fibonacci sequence to formulate a conjecture on the $t$-metric Mahler measures. If the conjecture is true, we prove that it is best possible and that it implies the the existence of rational numbers with as many exceptional points as we like. Finally, with some computational assistance, we resolve various special cases of the conjecture that constitute improvements to earlier results.

Many machine learning problems can be characterized by mutual contamination models. In these problems, one observes several random samples from different convex combinations of a set of unknown base distributions. It is of interest to decontaminate mutual contamination models, i.e., to recover the base distributions either exactly or up to a permutation. This paper considers the general setting where the base distributions are defined on arbitrary probability spaces. We examine the decontamination problem in two mutual contamination models that describe popular machine learning tasks: recovering the base distributions up to a permutation in a mixed membership model, and recovering the base distributions exactly in a partial label model for classification. We give necessary and sufficient conditions for identifiability of both mutual contamination models, algorithms for both problems in the infinite and finite sample cases, and introduce novel proof techniques based on affine geometry.