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Paired single-cell sequencing technologies enable the simultaneous measurement of complementary modalities of molecular data at single-cell resolution. Along with the advances in these technologies, many methods based on variational autoencoders have been developed to integrate these data. However, these methods do not explicitly incorporate prior biological relationships between the data modalities, which could significantly enhance modeling and interpretation. We propose a novel probabilistic learning framework that explicitly incorporates conditional independence relationships between multi-modal data as a directed acyclic graph using a generalized hierarchical variational autoencoder. We demonstrate the versatility of our framework across various applications pertinent to single-cell multi-omics data integration. These include the isolation of common and distinct information from different modalities, modality-specific differential analysis, and integrated cell clustering. We anticipate that the proposed framework can facilitate the construction of highly flexible graphical models that can capture the complexities of biological hypotheses and unravel the connections between different biological data types, such as different modalities of paired single-cell multi-omics data. The implementation of the proposed framework can be found in the repository https://github.com/kuijjerlab/CAVACHON.

We discuss the process of Higgs boson production in $\gamma\gamma$ collider on noncommutative spacetime and compare the results with large extra dimension in KK graviton channel. Summing all KK mode on IR brane, the affections are in the same order by comparing noncommutatve model prediction. This process is completely forbidden in standard model on unitarity condition and bosonic distribution. In noncommutative theory, the effect is induced by the coordinates noncommutable relation, $[x^{\mu}, x^{\nu}]$ = $i\theta^{\mu\nu}$. Due to the constant background strength tensor does not contain any conserved quantum number, hence, this effort is indicated into particle power spectrum. Particle mass spectrum is corrected by radiational and anisotropic surroundings. The process of $\gamma\gamma\to H^{0}H^{0}$ restricts the unitarity condition in noncommutative field theory. Under power law conservation condition, the neutral Higgs mass below gauge boson resonance will produce a accelerated phenomena as the central energy is higher than $Z_{0}$ gauge boson creation scale. The effects generated from the vast light Higgs particles descend the power rate energy distribution as far as the ambient is under a balance. The fractional rate on polarized polars are very small embedded into the unpolarized surroundings depend on background electric field couplings., Comment: 15 pages, 19 figures

If the twist $Poincar\acute{e}$ transformation is imposed on the noncommutative spacetime, then $Lorentz$ invariance cannot be applied on QFT. To data, noncommutative theory is one of the best candidates to modify $Lorentz$ transformation. In this paper, we argue parity violation under the process of $e^{+}e^{-}\to\gamma \gamma$ and make a detailed analysis of the difference behavior of each helicity state on noncommutative spacetime. The effect arises from the production of spin and magnetic fields. We check the energy momentum conservation for all used couplings and discover that if the electric field changes particle energy spectrum, there is no symmetry violation as the field produces a longitudinal state on the finial triple boson couplings., Comment: 8 pages, 7 figures

In this paper, we consider a time dependent module field on spacetime extension without modifying commutative relation on noncommutative quantum plane. The significant idea is that $Lorentz$ symmetry is conserved in module and unmodule coordinate. We focus on the redefinition of spacetime structure without considering noncommutative bosonic gas in deforming the product between fields. Which the null vector is a vector on orthogonal $D$ dimensional $Hilbert$ spacetime. In $Riemann$ geometry, the equation of motion is deformed from an induced rotation. Particle field survives on the state composed by two theoretical assumed $null$ vectors, one is commutative, another is anticommutative. In the point of view, neutrino and photon mass are produced by its shift, the rotated effect generates a horizon in redefining particle field., Comment: 10 pages

This article proposes a novel density estimation based algorithm for carrying out supervised machine learning. The proposed algorithm features O(n) time complexity for generating a classifier, where n is the number of sampling instances in the training dataset. This feature is highly desirable in contemporary applications that involve large and still growing databases. In comparison with the kernel density estimation based approaches, the mathe-matical fundamental behind the proposed algorithm is not based on the assump-tion that the number of training instances approaches infinite. As a result, a classifier generated with the proposed algorithm may deliver higher prediction accuracy than the kernel density estimation based classifier in some cases., Comment: Inclusion of a new "Remarks" section

In recent years, kernel density estimation has been exploited by computer scientists to model machine learning problems. The kernel density estimation based approaches are of interest due to the low time complexity of either O(n) or O(n*log(n)) for constructing a classifier, where n is the number of sampling instances. Concerning design of kernel density estimators, one essential issue is how fast the pointwise mean square error (MSE) and/or the integrated mean square error (IMSE) diminish as the number of sampling instances increases. In this article, it is shown that with the proposed kernel function it is feasible to make the pointwise MSE of the density estimator converge at O(n^-2/3) regardless of the dimension of the vector space, provided that the probability density function at the point of interest meets certain conditions., Comment: The new version includes an additional theorem, Theorem 3