arXiv subjects

Jiyuan Zhang

Publications and source records attributed to Jiyuan Zhang.

80 records · Page 5Linked to original sources

arXiv2019

Co-rank 1 projections and the randomised Horn problem

Let $\hat{\boldsymbol x}$ be a normalised standard complex Gaussian vector, and project an Hermitian matrix $A$ onto the hyperplane orthogonal to $\hat{\boldsymbol x}$. In a recent paper Faraut [Tunisian J. Math. \textbf{1} (2019), 585--606] has observed that the corresponding eigenvalue PDF has an almost identical structure to the eigenvalue PDF for the rank 1 perturbation $A + b \hat{\boldsymbol x} \hat{\boldsymbol x}^\dagger$, and asks for an explanation. We provide this by way of a common derivation involving the secular equations and associated Jacobians. This applies too in related setting, for example when $\hat{\boldsymbol x}$ is a real Gaussian and $A$ Hermitian, and also in a multiplicative setting $A U B U^\dagger$ where $A, B$ are fixed unitary matrices with $B$ a multiplicative rank 1 deviation from unity, and $U$ is a Haar distributed unitary matrix. Specifically, in each case there is a dual eigenvalue problem giving rise to a PDF of almost identical structure.

math-ph↗

arXiv2019

Memory-Attended Recurrent Network for Video Captioning

Typical techniques for video captioning follow the encoder-decoder framework, which can only focus on one source video being processed. A potential disadvantage of such design is that it cannot capture the multiple visual context information of a word appearing in more than one relevant videos in training data. To tackle this limitation, we propose the Memory-Attended Recurrent Network (MARN) for video captioning, in which a memory structure is designed to explore the full-spectrum correspondence between a word and its various similar visual contexts across videos in training data. Thus, our model is able to achieve a more comprehensive understanding for each word and yield higher captioning quality. Furthermore, the built memory structure enables our method to model the compatibility between adjacent words explicitly instead of asking the model to learn implicitly, as most existing models do. Extensive validation on two real-word datasets demonstrates that our MARN consistently outperforms state-of-the-art methods.

cs.CV↗

arXiv2018

Parametrising correlation matrices

Correlation matrices are the sub-class of positive definite real matrices with all entries on the diagonal equal to unity. Earlier work has exhibited a parametrisation of the corresponding Cholesky factorisation in terms of partial correlations, and also in terms of hyperspherical co-ordinates. We show how the two are relating, starting from the definition of the partial correlations in terms of the Schur complement. We extend this to the generalisation of correlation matrices to the cases of complex and quaternion entries. As in the real case, we show how the hyperspherical parametrisation leads naturally to a distribution on the space of correlation matrices $\{R\}$ with probability density function proportional to $( \det R)^a$. For certain $a$, a construction of random correlation matrices realising this distribution is given in terms of rectangular standard Gaussian matrices.

math.ST↗

arXiv2018

High Performance Zero-Memory Overhead Direct Convolutions

The computation of convolution layers in deep neural networks typically rely on high performance routines that trade space for time by using additional memory (either for packing purposes or required as part of the algorithm) to improve performance. The problems with such an approach are two-fold. First, these routines incur additional memory overhead which reduces the overall size of the network that can fit on embedded devices with limited memory capacity. Second, these high performance routines were not optimized for performing convolution, which means that the performance obtained is usually less than conventionally expected. In this paper, we demonstrate that direct convolution, when implemented correctly, eliminates all memory overhead, and yields performance that is between 10% to 400% times better than existing high performance implementations of convolution layers on conventional and embedded CPU architectures. We also show that a high performance direct convolution exhibits better scaling performance, i.e. suffers less performance drop, when increasing the number of threads.

cs.LG↗

arXiv2018

Chinese Poetry Generation with Flexible Styles

Research has shown that sequence-to-sequence neural models, particularly those with the attention mechanism, can successfully generate classical Chinese poems. However, neural models are not capable of generating poems that match specific styles, such as the impulsive style of Li Bai, a famous poet in the Tang Dynasty. This work proposes a memory-augmented neural model to enable the generation of style-specific poetry. The key idea is a memory structure that stores how poems with a desired style were generated by humans, and uses similar fragments to adjust the generation. We demonstrate that the proposed algorithm generates poems with flexible styles, including styles of a particular era and an individual poet.

cs.CL↗

arXiv2018

Volumes and distributions for random unimodular complex and quaternion lattices

Two themes associated with invariant measures on the matrix groups ${\rm SL}_N(\mathbb F)$, with $\mathbb F = \mathbb R, \mathbb C$ or $\mathbb H$, and their corresponding lattices parametrised by ${\rm SL}_N(\mathbb F)/{\rm SL}_N(\mathbb O)$, $\mathbb O$ being an appropriate Euclidean ring of integers, are considered. The first is the computation of the volume of the subset of ${\rm SL}_N(\mathbb F)$ with bounded 2-norm or Frobenius norm. Key here is the decomposition of measure in terms of the singular values. The form of the volume, for large values of the bound, is relevant to asymptotic counting problems in ${\rm SL}_N(\mathcal O)$. The second is the problem of lattice reduction in the case $N=2$. A unified proof of the validity of the appropriate analogue of the Lagrange--Gauss algorithm for computing the shortest basis is given. A decomposition of measure corresponding to the QR decomposition is used to specify the invariant measure in the coordinates of the shortest basis vectors. With $\mathbb F = \mathbb C$ this allows for the exact computation of the PDF of the first minimum (for $\mathcal O = \mathbb Z[i]$ and $\mathbb Z[(1+\sqrt{-3})/2]$), and the PDF of the second minimum and that of the angle between the minimal basis vectors (for $\mathcal O = \mathbb Z[i]$). It also encodes the specification of fundamental domains of the corresponding quotient spaces. Integration over the latter gives rise to certain number theoretic constants, which are also present in the asymptotic forms of the PDFs of the lengths of the shortest basis vectors. Siegel's mean value gives an alternative method to compute the arithmetic constants, allowing in particular the computation of the leading form of the PDF of the first minimum for $\mathbb F = \mathbb H$ and $\mathcal O$ the Hurwitz integers, for which direct integration was not possible.

math.NT↗

arXiv2018

Lyapunov exponents for some isotropic random matrix ensembles

A random matrix with rows distributed as a function of their length is said to be isotropic. When these distributions are Gaussian, beta type I, or beta type II, previous work has, from the viewpoint of integral geometry, obtained the explicit form of the distribution of the determinant. We use these result to evaluate the sum of the Lyapunov spectrum of the corresponding random matrix product, and we further give explicit expressions for the largest Lyapunov exponent. Generalisations to the case of complex or quaternion entries are also given. For standard Gaussian matrices $X$, the full Lyapunov spectrum for products of random matrices $I_N + {1 \over c} X$ is computed in terms of a generalised hypergeometric function in general, and in terms of a single single integral involving a modified Bessel function for the largest Lyapunov exponent.

math.PR↗

arXiv2017

Flexible and Creative Chinese Poetry Generation Using Neural Memory

It has been shown that Chinese poems can be successfully generated by sequence-to-sequence neural models, particularly with the attention mechanism. A potential problem of this approach, however, is that neural models can only learn abstract rules, while poem generation is a highly creative process that involves not only rules but also innovations for which pure statistical models are not appropriate in principle. This work proposes a memory-augmented neural model for Chinese poem generation, where the neural model and the augmented memory work together to balance the requirements of linguistic accordance and aesthetic innovation, leading to innovative generations that are still rule-compliant. In addition, it is found that the memory mechanism provides interesting flexibility that can be used to generate poems with different styles.

cs.AI↗