A.2 Training Settings of T eacher We provide training settings of the teacher w.r.t. In practice, we do not optimize the student and the generator via the plain losses in Eq. 4 and Eq. 6, Number of steps for pretraining G, δ: the bound in Eqs. A.4 Generator Architectures In Table 8, we show different architectures of the generator w.r.t. ResNetBlockY are provided in Table 9. ConvBlockX(c This is because the "uncond" generator has learned to jump "sum" generator enables stable training of our model and gives the best accuracy and crossentropy The "cat" generator only yields good results at "uncond" generator does not encounter any problem with MAD to learn faster than the "cat" generator. An important question is "What is a reasonable upper bound

The proliferation of interconnected devices in the Internet of Things (IoT) has led to an exponential increase in data, commonly known as Big IoT Data. Efficient retrieval of this heterogeneous data demands a robust indexing mechanism for effective organization. However, a significant challenge remains: the overlap in data space partitions during index construction. This overlap increases node access during search and retrieval, resulting in higher resource consumption, performance bottlenecks, and impedes system scalability. To address this issue, we propose three innovative heuristics designed to quantify and strategically reduce data space partition overlap. The volume-based method (VBM) offers a detailed assessment by calculating the intersection volume between partitions, providing deeper insights into spatial relationships. The distance-based method (DBM) enhances efficiency by using the distance between partition centers and radii to evaluate overlap, offering a streamlined yet accurate approach. Finally, the object-based method (OBM) provides a practical solution by counting objects across multiple partitions, delivering an intuitive understanding of data space dynamics. Experimental results demonstrate the effectiveness of these methods in reducing search time, underscoring their potential to improve data space partitioning and enhance overall system performance.

Many algorithms require discriminative boundaries, such as separating hyperplanes or hyperballs, or are specifically designed to work on spherical data. By applying inversive geometry, we show that the two discriminative boundaries can be used interchangeably, and that general Euclidean data can be transformed into spherical data, whenever a change in point distances is acceptable. We provide explicit formulae to embed general Euclidean data into spherical data and to unembed it back. We further show a duality between hyperspherical caps, i.e., the volume created by a separating hyperplane on spherical data, and hyperballs and provide explicit formulae to map between the two. We further provide equations to translate inner products and Euclidean distances between the two spaces, to avoid explicit embedding and unembedding. We also provide a method to enforce projections of the general Euclidean space onto hemi-hyperspheres and propose an intrinsic dimensionality based method to obtain "all-purpose" parameters. To show the usefulness of the cap-ball-duality, we discuss example applications in machine learning and vector similarity search.

The highly influential framework of conceptual spaces provides a geometric way of representing knowledge. Instances are represented by points in a high-dimensional space and concepts are represented by regions in this space. In this article, we extend our recent mathematical formalization of this framework by providing quantitative mathematical definitions for measuring relations between concepts: We develop formal ways for computing concept size, subsethood, implication, similarity, and betweenness. This considerably increases the representational capabilities of our formalization and makes it the most thorough and comprehensive formalization of conceptual spaces developed so far.

Screening is an effective technique for speeding up the training process of a sparse learning model by removing the features that are guaranteed to be inactive the process. In this paper, we present a efficient screening technique for sparse support vector machine based on variational inequality. The technique is both efficient and safe.