On two conjectural supercongruences of Apagodu and Zeilberger

Let the numbers $\alpha_n,\beta_n$ and $\gamma_n$ denote \begin{align*} \alpha_n=\sum_{k=0}^{n-1}{2k\choose k},\quad \beta_n=\sum_{k=0}^{n-1}{2k\choose k}\frac{1}{k+1}\quad\text{and}\quad \gamma_n=\sum_{k=0}^{n-1}{2k\choose k}\frac{3k+2}{k+1}, \end{align*} respectively. We prove that for any prime $p\ge 5$ and positive integer $n$ \begin{align*} \alpha_{np}&\equiv \left(\frac{p}{3}\right) \alpha_n \pmod{p^2},\\ \beta_{np}&\equiv \begin{cases} \displaystyle \beta_n \pmod{p^2},\quad &\text{if $p\equiv 1\pmod{3}$},\\ -\gamma_n \pmod{p^2},\quad &\text{if $p\equiv 2\pmod{3}$}, \end{cases} \end{align*} where $\left(\frac{\cdot}{p}\right)$ denotes the Legendre symbol. These two supercongruences were recently conjectured by Apagodu and Zeilberger.Comment: to appear in J. Difference Equ. Appl. This version is a bit different from the final version for publicatio

Proof of some divisibility results on sums involving binomial coefficients

By using the Rodriguez-Villegas-Mortenson supercongruences, we prove four supercongruences on sums involving binomial coefficients, which were originally conjectured by Sun. We also confirm a related conjecture of Guo on integer-valued polynomials.Comment: 6 page

Congruences on sums of super Catalan numbers

In this paper, we prove two congruences on the double sums of the super Catalan numbers (named by Gessel), which were recently conjectured by Apagodu.Comment: 8 page

On van Hamme's (A.2) and (H.2) supercongruences

In 1997, van Hamme conjectured 13 Ramanujan-type supercongruences labeled (A.2)--(M.2). Using some combinatorial identities discovered by Sigma, we extend (A.2) and (H.2) to supercongruences modulo $p^4$ for primes $p\equiv 3\pmod{4}$, which appear to be new.Comment: 9 page

Supercongruences for the (p−1)(p-1)th Ap\'ery number

In this paper, we prove two conjectural supercongruences on the $(p-1)$th Ap\'ery number, which were recently proposed by Z.-H. Sun.Comment: 9 page

Video (GIF) Sentiment Analysis using Large-Scale Mid-Level Ontology

With faster connection speed, Internet users are now making social network a huge reservoir of texts, images and video clips (GIF). Sentiment analysis for such online platform can be used to predict political elections, evaluates economic indicators and so on. However, GIF sentiment analysis is quite challenging, not only because it hinges on spatio-temporal visual contentabstraction, but also for the relationship between such abstraction and final sentiment remains unknown.In this paper, we dedicated to find out such relationship.We proposed a SentiPairSequence basedspatiotemporal visual sentiment ontology, which forms the midlevel representations for GIFsentiment. The establishment process of SentiPair contains two steps. First, we construct the Synset Forest to define the semantic tree structure of visual sentiment label elements. Then, through theSynset Forest, we organically select and combine sentiment label elements to form a mid-level visual sentiment representation. Our experiments indicate that SentiPair outperforms other competing mid-level attributes. Using SentiPair, our analysis frameworkcan achieve satisfying prediction accuracy (72.6%). We also opened ourdataset (GSO-2015) to the research community. GSO-2015 contains more than 6,000 manually annotated GIFs out of more than 40,000 candidates. Each is labeled with both sentiment and SentiPair Sequence

Multi-Stage Variational Auto-Encoders for Coarse-to-Fine Image Generation

Variational auto-encoder (VAE) is a powerful unsupervised learning framework for image generation. One drawback of VAE is that it generates blurry images due to its Gaussianity assumption and thus L2 loss. To allow the generation of high quality images by VAE, we increase the capacity of decoder network by employing residual blocks and skip connections, which also enable efficient optimization. To overcome the limitation of L2 loss, we propose to generate images in a multi-stage manner from coarse to fine. In the simplest case, the proposed multi-stage VAE divides the decoder into two components in which the second component generates refined images based on the course images generated by the first component. Since the second component is independent of the VAE model, it can employ other loss functions beyond the L2 loss and different model architectures. The proposed framework can be easily generalized to contain more than two components. Experiment results on the MNIST and CelebA datasets demonstrate that the proposed multi-stage VAE can generate sharper images as compared to those from the original VAE

Proof of a congruence on sums of powers of qq-binomial coefficients

We prove that, if $m,n\geqslant 1$ and $a_1,\ldots,a_m$ are nonnegative integers, then \begin{align*} \frac{[a_1+\cdots+a_m+1]!}{[a_1]!\ldots[a_m]!}\sum^{n-1}_{h=0}q^h\prod_{i=1}^m{h\brack a_i} \equiv 0\pmod{[n]}, \end{align*} where $[n]=\frac{1-q^n}{1-q}$, $[n]!=[n][n-1]\cdots[1]$, and ${a\brack b}=\prod_{k=1}^b\frac{1-q^{a-k+1}}{1-q^k}$. The $a_1=\cdots=a_m$ case confirms a recent conjecture of Z.-W. Sun. We also show that, if $p>\max\{a,b\}$ is a prime, then \begin{align*} \frac{[a+b+1]!}{[a]![b]!}\sum_{h=0}^{p-1}q^h{h\brack a}{h\brack b} \equiv (-1)^{a-b} q^{ab-{a\choose 2}-{b\choose 2}}[p]\pmod{[p]^2}. \end{align*}Comment: 5 page

Scalable Incremental Nonconvex Optimization Approach for Phase Retrieval

We aim to find a solution $\bm{x}\in\mathbb{C}^n$ to a system of quadratic equations of the form $b_i=\lvert\bm{a}_i^*\bm{x}\rvert^2$, $i=1,2,\ldots,m$, e.g., the well-known NP-hard phase retrieval problem. As opposed to recently proposed state-of-the-art nonconvex methods, we revert to the semidefinite relaxation (SDR) PhaseLift convex formulation and propose a successive and incremental nonconvex optimization algorithm, termed as \texttt{IncrePR}, to indirectly minimize the resulting convex problem on the cone of positive semidefinite matrices. Our proposed method overcomes the excessive computational cost of typical SDP solvers as well as the need of a good initialization for typical nonconvex methods. For Gaussian measurements, which is usually needed for provable convergence of nonconvex methods, \texttt{IncrePR} with restart strategy outperforms state-of-the-art nonconvex solvers with a sharper phase transition of perfect recovery and typical convex solvers in terms of computational cost and storage. For more challenging structured (non-Gaussian) measurements often occurred in real applications, such as transmission matrix and oversampling Fourier transform, \texttt{IncrePR} with several restarts can be used to find a good initial guess. With further refinement by local nonconvex solvers, one can achieve a better solution than that obtained by applying nonconvex solvers directly when the number of measurements is relatively small. Extensive numerical tests are performed to demonstrate the effectiveness of the proposed method.Comment: 20 pages, 25 figure

Proof of a conjecture of Z.-W. Sun on the divisibility of a triple sum

The numbers $R_n$ and $W_n$ are defined as \begin{align*} R_n=\sum_{k=0}^{n}{n+k\choose 2k}{2k\choose k}\frac{1}{2k-1},\ \text{and}\ W_n=\sum_{k=0}^{n}{n+k\choose 2k}{2k\choose k}\frac{3}{2k-3}. \end{align*} We prove that, for any positive integer $n$ and odd prime $p$, there hold \begin{align*} \sum_{k=0}^{n-1}(2k+1)R_k^2 &\equiv 0 \pmod{n}, \\ \sum_{k=0}^{p-1}(2k+1)R_k^2 &\equiv 4p(-1)^{\frac{p-1}{2}} -p^2 \pmod{p^3}, \\ 9\sum_{k=0}^{n-1}(2k+1)W_k^2 &\equiv 0 \pmod{n}, \\ \sum_{k=0}^{p-1}(2k+1)W_k^2 &\equiv 12p(-1)^{\frac{p-1}{2}}-17p^2 \pmod{p^3}, \quad\text{if $p>3$.} \end{align*} The first two congruences were originally conjectured by Z.-W. Sun. Our proof is based on the multi-variable Zeilberger algorithm and the following observation: $${2n\choose n}{n\choose k}{m\choose k}{k\choose m-n}\equiv 0\pmod{{2k\choose k}{2m-2k\choose m-k}},$$ where $0\leqslant k\leqslant n\leqslant m \leqslant 2n$.Comment: 18 page