Marvel: A Data-Centric Approach for Mapping Deep Learning Operators on Spatial Accelerators

A spatial accelerator’s efficiency depends heavily on both its mapper and cost models to generate optimized mappings for various operators of DNN models. However, existing cost models lack a formal boundary over their input programs (operators) for accurate and tractable cost analysis of the mappings, and this results in adaptability challenges to the cost models for new operators. We consider the recently introduced Maestro Data-Centric (MDC) notation and its analytical cost model to address this challenge because any mapping expressed in the notation is precisely analyzable using the MDC’s cost model. In this article, we characterize the set of input operators and their mappings expressed in the MDC notation by introducing a set of conformability rules . The outcome of these rules is that any loop nest that is perfectly nested with affine tensor subscripts and without conditionals is conformable to the MDC notation. A majority of the primitive operators in deep learning are such loop nests. In addition, our rules enable us to automatically translate a mapping expressed in the loop nest form to MDC notation and use the MDC’s cost model to guide upstream mappers. Our conformability rules over the input operators result in a structured mapping space of the operators, which enables us to introduce a mapper based on our decoupled off-chip/on-chip approach to accelerate mapping space exploration. Our mapper decomposes the original higher-dimensional mapping space of operators into two lower-dimensional off-chip and on-chip subspaces and then optimizes the off-chip subspace followed by the on-chip subspace. We implemented our overall approach in a tool called Marvel , and a benefit of our approach is that it applies to any operator conformable with the MDC notation. We evaluated Marvel over major DNN operators and compared it with past optimizers.

Elements of ∞-Category Theory

The language of ∞-categories provides an insightful new way of expressing many results in higher-dimensional mathematics but can be challenging for the uninitiated. To explain what exactly an ∞-category is requires various technical models, raising the question of how they might be compared. To overcome this, a model-independent approach is desired, so that theorems proven with any model would apply to them all. This text develops the theory of ∞-categories from first principles in a model-independent fashion using the axiomatic framework of an ∞-cosmos, the universe in which ∞-categories live as objects. An ∞-cosmos is a fertile setting for the formal category theory of ∞-categories, and in this way the foundational proofs in ∞-category theory closely resemble the classical foundations of ordinary category theory. Equipped with exercises and appendices with background material, this first introduction is meant for students and researchers who have a strong foundation in classical 1-category theory.

USING BLOCK PSEUDOREVERSION IN SEQUENTIAL DATA PROCESSING

Предложен алгоритм последовательной обработки данных на основе блочного псевдообращения матриц полного столбцового ранга. Показывается, что формула блочного псевдообращения, лежащая в основе алгоритма, является обобщением одного шага алгоритма Гревиля псевдообращения в невырожденном случае и потому может быть использована для обобщения метода нахождения весов нейросетевой функции LSHDI (linear solutions to higher dimensional interlayer networks), основанного на алгоритме Гревиля. Представленный алгоритм на каждом этапе использует найденные на предыдущих этапах псевдообратные к блокам матрицы и, следовательно, позволяет сократить вычисления не только за счет работы с матрицами меньшего размера, но и за счет повторного использования уже найденной информации. Приводятся примеры применения алгоритма для восстановления искаженных работой фильтра (шума) одномерных сигналов и двумерных сигналов (изображений). Рассматриваются случаи, когда фильтр является статическим, но на практике встречаются ситуации, когда матрица фильтра меняется с течением времени. Описанный алгоритм позволяет непосредственно в процессе получения входного сигнала перестраивать псевдообратную матрицу с учетом изменения одного или нескольких блоков матрицы фильтра, и потому алгоритм может быть использован и в случае зависящих от времени параметров фильтра (шума). Кроме того, как показывают вычислительные эксперименты, формула блочного псевдообращения, на которой основан описываемый алгоритм, хорошо работает и в случае плохо обусловленных матриц, что часто встречается на практике The paper proposes an algorithm for sequential data processing based on block pseudoinverse of full column rank matrixes. It is shown that the block pseudoinverse formula underlying the algorithm is a generalization of one step of the Greville’s pseudoinverse algorithm in the nonsingular case and can also be used as a generalization for finding weights of neural network function in the LSHDI algorithm (linear solutions to higher dimensional interlayer networks). The presented algorithm uses the pseudoinversed matrixes found at each step, and therefore allows one to reduce the computations not only by working with matrixes of smaller size but also by reusing the already found information. Examples of application of the algorithm for signal and image reconstruction are given. The article deals with cases where noise is static but the algorithm is similarly well suited to dynamically changing noises, allowing one to process input data in blocks on the fly, depending on changes. The block pseudoreverse formula, on which the described algorithm is based, works well in the case of ill-conditioned matrixes, which is often encountered in practice

Higher-Dimensional Regular Reissner–Nordström Black Holes Associated with Linear Electrodynamics

Following the interpretation of matter source that the energy-momentum tensor of anisotropic fluid can be dealt with effectively as the energy-momentum tensor of perfect fluid plus linear (Maxwell) electromagnetic field, we obtain the regular higher-dimensional Reissner–Nordström (Tangherlini–RN) solution by starting with the noncommutative geometry-inspired Schwarzschild solution. Using the boundary conditions that connect the noncommutative Schwarzschild solution in the interior of the charged perfect fluid sphere to the Tangherlini–RN solution in the exterior of the sphere, we find that the interior structure can be reflected by an exterior parameter, the charge-to-mass ratio. Moreover, we investigate the stability of the boundary under mass perturbation and indicate that the new interpretation imposes a rigid restriction upon the charge-to-mass ratio. This restriction, in turn, permits a stable noncommutative black hole only in the 4-dimensional spacetime.

Mixed Position and Twist Space Synthesis of 3R Chains

Mixed-position kinematic synthesis is used to not only reach a certain number of precision positions, but also impose certain instantaneous motion conditions at those positions. In the traditional approach, one end-effector twist is defined at each precision position in order to achieve better guidance of the end-effector along a desired trajectory. For one-degree-of-freedom linkages, that suffices to fully specify the trajectory locally. However, for systems with a higher number of degrees of freedom, such as robotic systems, it is possible to specify a complete higher-dimensional subspace of potential twists at particular positions. In this work, we focus on the 3R serial chain. We study the three-dimensional subspaces of twists that can be defined and set the mixed-position equations to synthesize the chain. The number and type of twist systems that a chain can generate depend on the topology of the chain; we find that the spatial 3R chain can generate seven different fully defined twist systems. Finally, examples of synthesis with several fully defined and partially defined twist spaces are presented. We show that it is possible to synthesize 3R chains for feasible subspaces of different types. This allows a complete definition of potential motions at particular positions, which could be used for the design of precise interaction with contact surfaces.

Asymptotic quasinormal frequencies of different spin fields in d-dimensional spherically-symmetric black holes

While Hod's conjecture is demonstrably restrictive, the link he observed between black hole (BH) area quantisation and the large overtone ($n$) limit of quasinormal frequencies (QNFs) motivated intense scrutiny of the regime, from which an improved understanding of asymptotic quasinormal frequencies (aQNFs) emerged. A further outcome was the development of the ``monodromy technique", which exploits an anti-Stokes line analysis to extract physical solutions from the complex plane. Here, we use the monodromy technique to validate extant aQNF expressions for perturbations of integer spin, and provide new results for the aQNFs of half-integer spins within higher-dimensional Schwarzschild, Reissner-Nordstr{\"o}m, and Schwarzschild (anti-)de Sitter BH spacetimes. Bar the Schwarzschild anti-de Sitter case, the spin-1/2 aQNFs are purely imaginary; the spin-3/2 aQNFs resemble spin-1/2 aQNFs in Schwarzschild and Schwarzschild de Sitter BHs, but match the gravitational perturbations for most others. Particularly for Schwarzschild, extremal Reissner-Nordstr{\"o}m, and several Schwarzschild de Sitter cases, the application of $n \rightarrow \infty$ generally fixes $\mathbb{R}e \{ \omega \}$ and allows for the unbounded growth of $\mathbb{I}m \{ \omega \}$ in fixed quantities.

Efficient simulation of 3D reaction-diffusion in models of neurons and networks

Neuronal activity is the result of both the electrophysiology and chemophysiology. A neuron can be well represented for the purposes of electrophysiological simulation as a tree composed of connected cylinders. This representation is also apt for 1D simulations of their chemophysiology, provided the spatial scale is larger than the diameter of the cylinders and there is radial symmetry. Higher dimensional simulation is necessary to accurately capture the dynamics when these criteria are not met, such as with wave curvature, spines, or diffusion near the soma. We have developed a solution to enable efficient finite volume method simulation of reaction-diffusion kinetics in intracellular 3D regions in neuron and network models and provide an implementation within the NEURON simulator. An accelerated version of the CTNG 3D reconstruction algorithm transforms morphologies suitable for ion-channel based simulations into consistent 3D voxelized regions. Kinetics are then solved using a parallel algorithm based on Douglas-Gunn that handles the irregular 3D geometry of a neuron; these kinetics are coupled to NEURON's 1D mechanisms for ion channels, synapses, etc. The 3D domain may cover the entire cell or selected regions of interest. Simulations with dendritic spines and of the soma reveal details of dynamics that would be missed in a pure 1D simulation. We describe and validate the methods and discuss their performance.