instruction:
In this task, you are given the abstract of a research paper. Your task is to generate a summary of this abstract. Your summary should not be very short, but it's better if it's not more than 30 words.
question:
We consider a simple and overarching representation for permutation-invariant functions of sequences (or set functions). Our approach, which we call Janossy pooling, expresses a permutation-invariant function as the average of a permutation-sensitive function applied to all reorderings of the input sequence. This allows us to leverage the rich and mature literature on permutation-sensitive functions to construct novel and flexible permutation-invariant functions. If carried out naively, Janossy pooling can be computationally prohibitive. To allow computational tractability, we consider three kinds of approximations: canonical orderings of sequences, functions with k-order interactions, and stochastic optimization algorithms with random permutations. Our framework unifies a variety of existing work in the literature, and suggests possible modeling and algorithmic extensions. We explore a few in our experiments, which demonstrate improved performance over current state-of-the-art methods.
answer:
We propose Janossy pooling, a method for learning deep permutation invariant functions designed to exploit relationships within the input sequence and tractable inference strategies such as a stochastic optimization procedure we call piSGD


question:
Sometimes SRS (Stereotactic Radio Surgery) requires using sphere packing on a Region of Interest (ROI) such as cancer to determine a treatment plan.   We have developed a sphere packing algorithm which packs non-intersecting spheres inside the ROI.   The region of interest in our case are those voxels which are identified as cancer tissues.   In this paper, we analyze the rotational invariant properties of  our sphere-packing algorithm which is based on distance transformations. Epsilon-Rotation invariant means the ability to arbitrary rotate the 3D ROI while keeping the volume properties remaining (almost) same within some limit of epsilon. The applied rotations produce spherical packing which remains highly correlated as we analyze the geometrically properties of sphere packing before and after the rotation of the volume data for the ROI. Our novel sphere packing algorithm has high degree of rotation invariance within the range of +/- epsilon. Our method used a shape descriptor derived from the values of the disjoint set of spheres form the distance-based sphere packing algorithm to extract the invariant descriptor from the ROI. We demonstrated by implementing these ideas using Slicer3D platform available for our research.   The data is based on sing MRI Stereotactic images. We presented several performance results on different benchmarks data of over 30 patients in Slicer3D platform.
answer:
Packing region of Interest (ROI) such as cancerous regions identified in 3D Volume Data, Packing spheres inside the ROI, rotating the ROI , measures of difference in sphere packing before and after the rotation.


question:
Variational inference (VI) methods and especially variational autoencoders (VAEs) specify scalable generative models that enjoy an intuitive connection to manifold learning --- with many default priors the posterior/likelihood pair $q(z|x)$/$p(x|z)$ can be viewed as an approximate homeomorphism (and its inverse) between the data manifold and a latent Euclidean space. However, these approximations are well-documented to become degenerate in training. Unless the subjective prior is carefully chosen, the topologies of the prior and data distributions often will not match.
 Conversely, diffusion maps (DM) automatically \textit{infer} the data topology and enjoy a rigorous connection to manifold learning, but do not scale easily or provide the inverse homeomorphism.
 In this paper, we propose \textbf{a)} a principled measure for recognizing the mismatch between data and latent distributions and \textbf{b)} a method that combines the advantages of variational inference and diffusion maps to learn a homeomorphic generative model. The measure, the \textit{locally bi-Lipschitz property}, is a sufficient condition for a homeomorphism and easy to compute and interpret. The method, the \textit{variational diffusion autoencoder} (VDAE), is a novel generative algorithm that first infers the topology of the data distribution, then models a diffusion random walk over the data. To achieve efficient computation in VDAEs, we use stochastic versions of both variational inference and manifold learning optimization. We prove approximation theoretic results for the dimension dependence of VDAEs, and that locally isotropic sampling in the latent space results in a random walk over the reconstructed manifold.
 Finally, we demonstrate the utility of our method on various real and synthetic datasets, and show that it exhibits performance superior to other generative models.
answer:

We combine variational inference and manifold learning (specifically VAEs and diffusion maps) to build a generative model based on a diffusion random walk on a data manifold; we generate samples by drawing from the walk's stationary distribution.