arXiv subjects

Lin F. Yang

Publications and source records attributed to Lin F. Yang.

96 records · Page 6Linked to original sources

arXiv2017

Online Factorization and Partition of Complex Networks From Random Walks

Finding the reduced-dimensional structure is critical to understanding complex networks. Existing approaches such as spectral clustering are applicable only when the full network is explicitly observed. In this paper, we focus on the online factorization and partition of implicit large-scale networks based on observations from an associated random walk. We formulate this into a nonconvex stochastic factorization problem and propose an efficient and scalable stochastic generalized Hebbian algorithm. The algorithm is able to process dependent state-transition data dynamically generated by the underlying network and learn a low-dimensional representation for each vertex. By applying a diffusion approximation analysis, we show that the continuous-time limiting process of the stochastic algorithm converges globally to the "principal components" of the Markov chain and achieves a nearly optimal sample complexity. Once given the learned low-dimensional representations, we further apply clustering techniques to recover the network partition. We show that when the associated Markov process is lumpable, one can recover the partition exactly with high probability. We apply the proposed approach to model the traffic flow of Manhattan as city-wide random walks. By using our algorithm to analyze the taxi trip data, we discover a latent partition of the Manhattan city that closely matches the traffic dynamics.

cs.LG↗

arXiv2017

Streaming Symmetric Norms via Measure Concentration

We characterize the streaming space complexity of every symmetric norm $l$ (a norm on $\mathbb{R}^n$ invariant under sign-flips and coordinate-permutations), by relating this space complexity to the measure-concentration characteristics of $l$. Specifically, we provide nearly matching upper and lower bounds on the space complexity of calculating a $(1\pmε)$-approximation to the norm of the stream, for every $0<ε\leq 1/2$. (The bounds match up to $poly(ε^{-1} \log n)$ factors.) We further extend those bounds to any large approximation ratio $D\geq 1.1$, showing that the decrease in space complexity is proportional to $D^2$, and that this factor the best possible. All of the bounds depend on the median of $l(x)$ when $x$ is drawn uniformly from the $l_2$ unit sphere. The same median governs many phenomena in high-dimensional spaces, such as large-deviation bounds and the critical dimension in Dvoretzky's Theorem. The family of symmetric norms contains several well-studied norms, such as all $l_p$~norms, and indeed we provide a new explanation for the disparity in space complexity between $p\le 2$ and $p>2$. In addition, we apply our general results to easily derive bounds for several norms that were not studied before in the streaming model, including the top-$k$ norm and the $k$-support norm, which was recently employed for machine learning tasks. Overall, these results make progress on two outstanding problems in the area of sublinear algorithms (Problems 5 and 30 in~\url{http://sublinear.info}).

cs.DS↗

arXiv2017

Clustering High Dimensional Dynamic Data Streams

We present data streaming algorithms for the $k$-median problem in high-dimensional dynamic geometric data streams, i.e. streams allowing both insertions and deletions of points from a discrete Euclidean space $\{1, 2, \ldots Δ\}^d$. Our algorithms use $k ε^{-2} poly(d \log Δ)$ space/time and maintain with high probability a small weighted set of points (a coreset) such that for every set of $k$ centers the cost of the coreset $(1+ε)$-approximates the cost of the streamed point set. We also provide algorithms that guarantee only positive weights in the coreset with additional logarithmic factors in the space and time complexities. We can use this positively-weighted coreset to compute a $(1+ε)$-approximation for the $k$-median problem by any efficient offline $k$-median algorithm. All previous algorithms for computing a $(1+ε)$-approximation for the $k$-median problem over dynamic data streams required space and time exponential in $d$. Our algorithms can be generalized to metric spaces of bounded doubling dimension.

cs.DS↗

arXiv2016

Streaming Space Complexity of Nearly All Functions of One Variable on Frequency Vectors

A central problem in the theory of algorithms for data streams is to determine which functions on a stream can be approximated in sublinear, and especially sub-polynomial or poly-logarithmic, space. Given a function $g$, we study the space complexity of approximating $\sum_{i=1}^n g(|f_i|)$, where $f\in\mathbb{Z}^n$ is the frequency vector of a turnstile stream. This is a generalization of the well-known frequency moments problem, and previous results apply only when $g$ is monotonic or has a special functional form. Our contribution is to give a condition such that, except for a narrow class of functions $g$, there is a space-efficient approximation algorithm for the sum if and only if $g$ satisfies the condition. The functions $g$ that we are able to characterize include all convex, concave, monotonic, polynomial, and trigonometric functions, among many others, and is the first such characterization for non-monotonic functions. Thus, for nearly all functions of one variable, we answer the open question from the celebrated paper of Alon, Matias and Szegedy (1996).

cs.DS↗

arXiv2015

Warmth Elevating the Depths: Shallower Voids with Warm Dark Matter

Warm dark matter (WDM) has been proposed as an alternative to cold dark matter (CDM), to resolve issues such as the apparent lack of satellites around the Milky Way. Even if WDM is not the answer to observational issues, it is essential to constrain the nature of the dark matter. The effect of WDM on haloes has been extensively studied, but the small-scale initial smoothing in WDM also affects the present-day cosmic web and voids. It suppresses the cosmic "sub-web" inside voids, and the formation of both void haloes and subvoids. In $N$-body simulations run with different assumed WDM masses, we identify voids with the ZOBOV algorithm, and cosmic-web components with the ORIGAMI algorithm. As dark-matter warmth increases (i.e., particle mass decreases), void density minima grow shallower, while void edges change little. Also, the number of subvoids decreases. The density field in voids is particularly insensitive to baryonic physics, so if void density profiles and minima could be measured observationally, they would offer a valuable probe of the nature of dark matter. Furthermore, filaments and walls become cleaner, as the substructures in between have been smoothed out; this leads to a clear, mid-range peak in the density PDF.

astro-ph.CO↗

arXiv2014

The Dark Matter Contribution to Galactic Diffuse Gamma Ray Emission

Observations of diffuse Galactic gamma ray emission (DGE) by the Fermi Large Area Telescope (LAT) allow a detailed study of cosmic rays and the interstellar medium. However, diffuse emission models of the inner Galaxy underpredict the Fermi-LAT data at energies above a few GeV and hint at possible non-astrophysical sources including dark matter (DM) annihilations or decays. We present a study of the possible emission components from DM using the high-resolution Via Lactea II N-body simulation of a Milky Way-sized DM halo. We generate full-sky maps of DM annihilation and decay signals that include modeling of the adiabatic contraction of the host density profile, Sommerfeld enhanced DM annihilations, $p$-wave annihilations, and decaying DM. We compare our results with the DGE models produced by the Fermi-LAT team over different sky regions, including the Galactic center, high Galactic latitudes, and the Galactic anti-center. This work provides possible templates to fit the observational data that includes the contribution of the subhalo population to DM gamma-ray emission, with the significance depending on the annihilation/decay channels and the Galactic regions being considered.

astro-ph.CO↗