Multiple Product Modulo Arbitrary Numbers

AbstractLetnbinary numbers of lengthnbe given. The Boolean function “Multiple Product”MPnasks for (some binary representation of ) the value of their product. It has been shown (K.-Y. Siu and V. Roychowdhury, On optimal depth threshold circuits for multiplication and related problems,SIAM J. Discrete Math.7, 285–292 (1994)) that this function can be computed in polynomial-size threshold circuits of depth 4. For many other arithmetic functions, circuits of depth 3 are known. They are mostly based on the fact that the value of the considered function modulo some prime numbers p can be computed easily in threshold circuits of depth 2. In this paper, we investigate the complexity of computingMPnmodulomby depth-2 threshold circuits. It turns out that for all but a few integersm, exponential size is required. In particular, it is shown that form∈{2, 4, 8}, polynomial-size circuits exist, form∈{3, 6, 12, 24}, the question remains open and in all other cases, exponential-size circuits are required. The result still holds if we allowmto grow withn

Why and When Can Deep -- but Not Shallow -- Networks Avoid the Curse of Dimensionality: a Review

The paper characterizes classes of functions for which deep learning can be exponentially better than shallow learning. Deep convolutional networks are a special case of these conditions, though weight sharing is not the main reason for their exponential advantage

Visual Cortical Mechanisms of Perceptual Grouping: Interacting Layers, Networks, Columns, and Maps

The visual cortex has a laminar organintion whose circuits form functional columns in cortical maps. How this laminar architecture supports visual percepts is not well understood. A neural model proposes how the laminar circuits of Vl and V2 generate perceptual groupings that maintain sensitivity to the contrasts and spatial organization of scenic cues.The model can decisively choose which groupings cohere and survive, even while balanced excitatory and inhibitory interactions preserve contrast-sensitive measures of local boundary likelihood or strength. In the model, excitatory inputs from LGN activate layers 4 and 6 of Vl. Layer 6 activates an on-center off-surround network of inputs to layer 4. Together these layer 4 inputs preserve analog sensitivity to LGN input contrast. Layer 4 cells excite pyramidal cells in layer 2/3 which activate monosynaptic long-range horizontal excitory connections between layer 2/3 pyramidal cells, and short-range disynaptic inhibitory connections mediated by smooth stellate cells. These interactions support inward perceptual grouping between two or more boundary inducetd, but not outward grouping from a single inducer. These lJO\UHlary signals feed back to layer 4 via the layer 6-to-4 on-center off-surround network. This folded feecdback joind cells in different layers into functional columnns while selecting winning groupings. Layer G in V1 also sends top-dlown signals to LGN using an on-center off-surround network, which suppresses LGN cells that do not receive feedback, while selecting, enhaneing, and synchronizing activity of those that do. The model is used to simulate psychophysical and neurophysiological data about perceptual grouping, including various Gestalt grouping laws.Air Force Office of Scientific Research (D0-0175), British Petroleum (BP 89A-1204); Defense Advanced Research Projects Agency and Office of Naval Research (N00014-92-J-4015); HNC Software (SC9-4-001), National Science Foundation (IRI-90-00530); Office of Naval Research (N00014-1-91-4100, N00014-95-1-0409, 0NR N00014-9510657

Memristive Computing

Memristive computing refers to the utilization of the memristor, the fourth fundamental passive circuit element, in computational tasks. The existence of the memristor was theoretically predicted in 1971 by Leon O. Chua, but experimentally validated only in 2008 by HP Labs. A memristor is essentially a nonvolatile nanoscale programmable resistor — indeed, memory resistor — whose resistance, or memristance to be precise, is changed by applying a voltage across, or current through, the device. Memristive computing is a new area of research, and many of its fundamental questions still remain open. For example, it is yet unclear which applications would benefit the most from the inherent nonlinear dynamics of memristors. In any case, these dynamics should be exploited to allow memristors to perform computation in a natural way instead of attempting to emulate existing technologies such as CMOS logic. Examples of such methods of computation presented in this thesis are memristive stateful logic operations, memristive multiplication based on the translinear principle, and the exploitation of nonlinear dynamics to construct chaotic memristive circuits. This thesis considers memristive computing at various levels of abstraction. The first part of the thesis analyses the physical properties and the current-voltage behaviour of a single device. The middle part presents memristor programming methods, and describes microcircuits for logic and analog operations. The final chapters discuss memristive computing in largescale applications. In particular, cellular neural networks, and associative memory architectures are proposed as applications that significantly benefit from memristive implementation. The work presents several new results on memristor modeling and programming, memristive logic, analog arithmetic operations on memristors, and applications of memristors. The main conclusion of this thesis is that memristive computing will be advantageous in large-scale, highly parallel mixed-mode processing architectures. This can be justified by the following two arguments. First, since processing can be performed directly within memristive memory architectures, the required circuitry, processing time, and possibly also power consumption can be reduced compared to a conventional CMOS implementation. Second, intrachip communication can be naturally implemented by a memristive crossbar structure.Siirretty Doriast

Measurement Quantum Cellular Automata and Anomalies in Floquet Codes

We investigate the evolution of quantum information under Pauli measurement circuits. We focus on the case of one- and two-dimensional systems, which are relevant to the recently introduced Floquet topological codes. We define local reversibility in context of measurement circuits, which allows us to treat finite depth measurement circuits on a similar footing to finite depth unitary circuits. In contrast to the unitary case, a finite depth locally reversible measurement circuit can implement a translation in one dimension. A locally reversible measurement circuit in two dimensions may also induce a flow of logical information along the boundary. We introduce "measurement quantum cellular automata" which unifies these ideas and define an index in one dimension to characterize the flow of logical operators. We find a $\mathbb{Z}_2$ bulk invariant for two-dimensional Floquet topological codes which indicates an obstruction to having a trivial boundary. We prove that the Hastings-Haah honeycomb code belongs to a class with such obstruction, which means that any boundary must have either nonlocal dynamics, period doubled, or admits anomalous boundary flow of quantum information.Comment: v2 changes: clarified the definition of "locally reversible measurement cycle" (LRMC), and added more examples of boundary circuits for the HH cod

Theory I: Why and When Can Deep Networks Avoid the Curse of Dimensionality?

[formerly titled "Why and When Can Deep – but Not Shallow – Networks Avoid the Curse of Dimensionality: a Review"] The paper reviews and extends an emerging body of theoretical results on deep learning including the conditions under which it can be exponentially better than shallow learning. A class of deep convolutional networks represent an important special case of these conditions, though weight sharing is not the main reason for their exponential advantage. Implications of a few key theorems are discussed, together with new results, open problems and conjectures.This work was supported by the Center for Brains, Minds and Machines (CBMM), funded by NSF STC award CCF – 1231216

Techniques for the realization of ultra- reliable spaceborne computer Final report

Bibliography and new techniques for use of error correction and redundancy to improve reliability of spaceborne computer