MIT Open Access Articles
https://hdl.handle.net/1721.1/49433
2020-01-21T00:08:43ZStochastic Airy semigroup through tridiagonal matrices
https://hdl.handle.net/1721.1/123482
Stochastic Airy semigroup through tridiagonal matrices
Gorin, Vadim; Shkolnikov, Mykhaylo
We determine the operator limit for large powers of random symmetric tridiagonal matrices as the size of the matrix grows. The result provides a novel expression in terms of functionals of Brownian motions for the Laplace transform of the Airy β process, which describes the largest eigenvalues in the β ensembles of random matrix theory. Another consequence is a Feynman-Kac formula for the stochastic Airy operator of Edelman-Sutton and Ramirez-Rider-Virag. As a side result, we find that the difference between the area underneath a standard Brownian excursion and one half of the integral of its squared local times is a Gaussian random variable. Keywords: Airy point process; Brownian bridge; Brownian excursion; Dumitriu–Edelman model; Feynman–Kac formula; Gaussian beta ensemble; intersection local time; moment method; path transformation; quantile transform; random matrix soft edge; random walk bridge; stochastic Airy operator; strong invariance principle; trace formula; Vervaat transform
2018-06-01T00:00:00ZInput-Output Distance Properties of Good Linear Codes
https://hdl.handle.net/1721.1/123481
Input-Output Distance Properties of Good Linear Codes
Hosseini Roozbehani, Hajir; Polyanskiy, Yury
Consider a linear code defined as a mapping between vector spaces of dimensions k and n. Let β* denote the minimal (relative) weight among all images of input vectors of full Hamming weight k. Operationally, β* characterizes the threshold for adversarial (erasure) noise beyond which decoder is guaranteed to produce estimate of k-input with 100% symbol error rate (SER). This paper studies the relation between β* and δ, the minimum distance of the code, which gives the threshold for 0 % SER. An optimal tradeoff between β* and δ is obtained (over large alphabets) and all linear codes achieving β*=1 are classified: they are repetition-like. More generally, a design criteria is proposed for codes with favorable graceful degradation properties. As an example, it is shown that in an overdetermined system of n homogeneous linear equations in k variables (over a field) it is always possible to satisfy some k-1 equations with non-zero assignments to every unknown, provided that any subset of k equations is linearly independent. This statement is true if and only if n ≥ 2k-1. Keywords: Linear codes; degradation; error correction codes; noise level; null space; hamming weight; error statistics
2018-06-01T00:00:00ZThe Sound of Pixels
https://hdl.handle.net/1721.1/123480
The Sound of Pixels
Zhao, Hang; Gan, Chuang; Rouditchenko, Andrew; Vondrick, Carl Martin; McDermott, Joshua Hartman; Torralba, Antonio
We introduce PixelPlayer, a system that, by leveraging large amounts of unlabeled videos, learns to locate image regions which produce sounds and separate the input sounds into a set of components that represents the sound from each pixel. Our approach capitalizes on the natural synchronization of the visual and audio modalities to learn models that jointly parse sounds and images, without requiring additional manual supervision. Experimental results on a newly collected MUSIC dataset show that our proposed Mix-and-Separate framework outperforms several baselines on source separation. Qualitative results suggest our model learns to ground sounds in vision, enabling applications such as independently adjusting the volume of sound sources. Keywords: Cross-modal learning; Sound separation and localization
2018-10-06T00:00:00ZOpen Vocabulary Scene Parsing
https://hdl.handle.net/1721.1/123479
Open Vocabulary Scene Parsing
Zhao, Hang; Puig Fernandez, Xavier; Zhou, Bolei; Fidler, Sanja; Torralba, Antonio
Recognizing arbitrary objects in the wild has been a challenging problem due to the limitations of existing classification models and datasets. In this paper, we propose a new task that aims at parsing scenes with a large and open vocabulary, and several evaluation metrics are explored for this problem. Our approach is a joint image pixel and word concept embeddings framework, where word concepts are connected by semantic relations. We validate the open vocabulary prediction ability of our framework on ADE20K dataset which covers a wide variety of scenes and objects. We further explore the trained joint embedding space to show its interpretability. Keywords: streaming media; vocabulary; training; semantics; predictive models; visualization
2017-12-25T00:00:00Z