On the Properties of Kullback-Leibler Divergence Between Multivariate Gaussian Distributions

Kullback-Leibler (KL) divergence is one of the most important divergence measures between probability distributions. In this paper, we prove several properties of KL divergence between multivariate Gaussian distributions. First, for any two $n$-dimensional Gaussian distributions $\mathcal{N}_1$ and $\mathcal{N}_2$, we give the supremum of $KL(\mathcal{N}_1||\mathcal{N}_2)$ when $KL(\mathcal{N}_2||\mathcal{N}_1)\leq \varepsilon\ (\varepsilon>0)$. For small $\varepsilon$, we show that the supremum is $\varepsilon + 2\varepsilon^{1.5} + O(\varepsilon^2)$. This quantifies the approximate symmetry of small KL divergence between Gaussians. We also find the infimum of $KL(\mathcal{N}_1||\mathcal{N}_2)$ when $KL(\mathcal{N}_2||\mathcal{N}_1)\geq M\ (M>0)$. We give the conditions when the supremum and infimum can be attained. Second, for any three $n$-dimensional Gaussians $\mathcal{N}_1$, $\mathcal{N}_2$, and $\mathcal{N}_3$, we find an upper bound of $KL(\mathcal{N}_1||\mathcal{N}_3)$ if $KL(\mathcal{N}_1||\mathcal{N}_2)\leq \varepsilon_1$ and $KL(\mathcal{N}_2||\mathcal{N}_3)\leq \varepsilon_2$ for $\varepsilon_1,\varepsilon_2\ge 0$. For small $\varepsilon_1$ and $\varepsilon_2$, we show the upper bound is $3\varepsilon_1+3\varepsilon_2+2\sqrt{\varepsilon_1\varepsilon_2}+o(\varepsilon_1)+o(\varepsilon_2)$. This reveals that KL divergence between Gaussians follows a relaxed triangle inequality. Importantly, all the bounds in the theorems presented in this paper are independent of the dimension $n$. Finally, We discuss the applications of our theorems in explaining counterintuitive phenomenon of flow-based model, deriving deep anomaly detection algorithm, and extending one-step robustness guarantee to multiple steps in safe reinforcement learning.Comment: arXiv admin note: text overlap with arXiv:2002.0332

Solar Flare Intensity Prediction With Machine Learning Models

We develop a mixed long short‐term memory (LSTM) regression model to predict the maximum solar flare intensity within a 24‐hr time window 0–24, 6–30, 12–36, and 24–48 hr ahead of time using 6, 12, 24, and 48 hr of data (predictors) for each Helioseismic and Magnetic Imager (HMI) Active Region Patch (HARP). The model makes use of (1) the Space‐Weather HMI Active Region Patch (SHARP) parameters as predictors and (2) the exact flare intensities instead of class labels recorded in the Geostationary Operational Environmental Satellites (GOES) data set, which serves as the source of the response variables. Compared to solar flare classification, the model offers us more detailed information about the exact maximum flux level, that is, intensity, for each occurrence of a flare. We also consider classification models built on top of the regression model and obtain better results in solar flare classifications as compared to Chen et al. (2019, https://doi.org/10.1029/2019SW002214). Our results suggest that the most efficient time period for predicting the solar activity is within 24 hr before the prediction time using the SHARP parameters and the LSTM model.Key PointsWe develop deep learning models to predict solar flare intensity values instead of flare classes from SHARP parameters in SDO/HMI data set directlyWe use time‐series information from both flaring time and nonflaring time in our modelAs opposed to solar flare classification, directly predicting solar flare intensity gives more detailed information about every occurrence of flares of each classPeer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/156246/2/swe21001_am.pdfhttp://deepblue.lib.umich.edu/bitstream/2027.42/156246/1/swe21001.pd

Semiquantum key distribution using initial states in only one basis without the classical user measuring

From the perspective of resource theory, it is interesting to achieve the same quantum task using as few quantum resources as possible. Semiquantum key distribution (SQKD), which allows a quantum user to share a confidential key with a classical user who prepares and operates qubits in only one basis, is an important example for studying this issue. To further limit the quantum resources used by users, in this paper, we constructed the first SQKD protocol which restricts the quantum user to prepare quantum states in only one basis and removes the classical user's measurement capability. Furthermore, we prove that the constructed protocol is unconditionally secure by deriving a key rate expression of the error rate in the asymptotic scenario. The work of this paper provides inspiration for achieving quantum superiority with minimal quantum resources.Comment: 13 pages, 3 figure

Soybean Breeding on Seed Composition Trait

Soybean is a most important crop providing edible oil and plant protein source for human beings, in addition to animal feed because of high protein and oil content. This review summarized the progresses in the QTL mapping, candidate gene cloning and functional analysis and also the regulation of soybean oil and seed storage protein accumulation. Furthermore, as soybean genome has been sequenced and released, prospects of multiple omics and advanced biotechnology should be combined and applied for further refine research and high-quality breeding

A novel type of cellular senescence that can be enhanced in mouse models and human tumor xenografts to suppress prostate tumorigenesis

Irreversible cell growth arrest, a process termed cellular senescence, is emerging as an intrinsic tumor suppressive mechanism. Oncogene-induced senescence is thought to be invariably preceded by hyperproliferation, aberrant replication, and activation of a DNA damage checkpoint response (DDR), rendering therapeutic enhancement of this process unsuitable for cancer treatment. We previously demonstrated in a mouse model of prostate cancer that inactivation of the tumor suppressor phosphatase and tensin homolog deleted on chromosome 10 (Pten) elicits a senescence response that opposes tumorigenesis. Here, we show that Pten-loss-induced cellular senescence (PICS) represents a senescence response that is distinct from oncogene-induced senescence and can be targeted for cancer therapy. Using mouse embryonic fibroblasts, we determined that PICS occurs rapidly after Pten inactivation, in the absence of cellular proliferation and DDR. Further, we found that PICS is associated with enhanced p53 translation. Consistent with these data, we showed that in mice p53-stabilizing drugs potentiated PICS and its tumor suppressive potential. Importantly, we demonstrated that pharmacological inhibition of PTEN drives senescence and inhibits tumorigenesis in vivo in a human xenograft model of prostate cancer. Taken together, our data identify a type of cellular senescence that can be triggered in nonproliferating cells in the absence of DNA damage, which we believe will be useful for developing a 'pro-senescence' approach for cancer prevention and therapy

KDM5B Is Essential for the Hyperactivation of PI3K/AKT Signaling in Prostate Tumorigenesis

KDM5B (lysine[K]-specific demethylase 5B) is frequently upregulated in various human cancers including prostate cancer. KDM5B controls H3K4me3/2 levels and regulates gene transcription and cell differentiation, yet the contributions of KDM5B to prostate cancer tumorigenesis remain unknown. In this study, we investigated the functional role of KDM5B in epigenetic dysregulation and prostate cancer progression in cultured cells and in mouse models of prostate epithelium–specific mutant Pten/Kdm5b. Kdm5b deficiency resulted in a significant delay in the onset of prostate cancer in Pten-null mice, whereas Kdm5b loss alone caused no morphologic abnormalities in mouse prostates. At 6 months of age, the prostate weight of Pten/Kdm5b mice was reduced by up to 70% compared with that of Pten mice. Pathologic analysis revealed Pten/Kdm5b mice displayed mild morphologic changes with hyperplasia in prostates, whereas age-matched Pten littermates developed high-grade prostatic intraepithelial neoplasia and prostate cancer. Mechanistically, KDM5B governed PI3K/AKT signaling in prostate cancer in vitro and in vivo. KDM5B directly bound the PIK3CA promoter, and KDM5B knockout resulted in a significant reduction of P110α and PIP3 levels and subsequent decrease in proliferation of human prostate cancer cells. Conversely, KDM5B overexpression resulted in increased PI3K/AKT signaling. Loss of Kdm5b abrogated the hyperactivation of AKT signaling by decreasing P110α/P85 levels in Pten/Kdm5b mice. Taken together, our findings reveal that KDM5B acts as a key regulator of PI3K/AKT signaling; they also support the concept that targeting KDM5B is a novel and effective therapeutic strategy against prostate cancer