Title: Resonator Networks, 2: Factorization Performance and Capacity Compared to Optimization-Based Methods
We develop theoretical foundations of resonator networks, a new type of recurrent neural network introduced in Frady, Kent, Olshausen, and Sommer (2020), a companion article in this issue, to solve a high-dimensional vector factorization problem arising in Vector Symbolic Architectures. Given a composite vector formed by the Hadamard product between a discrete set of high-dimensional vectors, a resonator network can efficiently decompose the composite into these factors. We compare the performance of resonator networks against optimization-based methods, including Alternating Least Squares and several gradient-based algorithms, showing that resonator networks are superior in several important ways. This advantage is achieved by leveraging a combination of nonlinear dynamics and searching in superposition, by which estimates of the correct solution are formed from a weighted superposition of all possible solutions. While the alternative methods also search in superposition, the dynamics of resonator networks allow them to strike a more effective balance between exploring the solution space and exploiting local information to drive the network toward probable solutions. Resonator networks are not guaranteed to converge, but within a particular regime they almost always do. In exchange for relaxing the guarantee of global convergence, resonator networks are dramatically more effective at finding factorizations than all alternative approaches considered. more »« less
Frady, E. Paxon; Kent, Spencer J.; Olshausen, Bruno A.; Sommer, Friedrich(
, Neural computation)
null
(Ed.)
The ability to encode and manipulate data structures with distributed neural representations could qualitatively enhance the capabilities of traditional neural networks by supporting rule-based symbolic reasoning, a central property of cognition. Here we show how this may be accomplished within the framework of Vector Symbolic Architectures (VSAs) (Plate, 1991; Gayler, 1998; Kanerva, 1996), whereby data structures are encoded by combining high-dimensional vectors with operations that together form an algebra on the space of distributed representations. In particular, we propose an efficient solution to a hard combinatorial search problem that arises when decoding elements of a VSA data structure: the factorization of products of multiple codevectors. Our proposed algorithm, called a resonator network, is a new type of recurrent neural network that interleaves VSA multiplication operations and pattern completion. We show in two examples—parsing of a tree-like data structure and parsing of a visual scene—how the factorization problem arises and how the resonator network can solve it. More broadly, resonator networks open the possibility of applying VSAs to myriad artificial intelligence problems in real-world domains. The companion article in this issue (Kent, Frady, Sommer, & Olshausen, 2020) presents a rigorous analysis and evaluation of the performance of resonator networks, showing it outperforms alternative approaches.
BACKGROUND Optical sensing devices measure the rich physical properties of an incident light beam, such as its power, polarization state, spectrum, and intensity distribution. Most conventional sensors, such as power meters, polarimeters, spectrometers, and cameras, are monofunctional and bulky. For example, classical Fourier-transform infrared spectrometers and polarimeters, which characterize the optical spectrum in the infrared and the polarization state of light, respectively, can occupy a considerable portion of an optical table. Over the past decade, the development of integrated sensing solutions by using miniaturized devices together with advanced machine-learning algorithms has accelerated rapidly, and optical sensing research has evolved into a highly interdisciplinary field that encompasses devices and materials engineering, condensed matter physics, and machine learning. To this end, future optical sensing technologies will benefit from innovations in device architecture, discoveries of new quantum materials, demonstrations of previously uncharacterized optical and optoelectronic phenomena, and rapid advances in the development of tailored machine-learning algorithms. ADVANCES Recently, a number of sensing and imaging demonstrations have emerged that differ substantially from conventional sensing schemes in the way that optical information is detected. A typical example is computational spectroscopy. In this new paradigm, a compact spectrometer first collectively captures the comprehensive spectral information of an incident light beam using multiple elements or a single element under different operational states and generates a high-dimensional photoresponse vector. An advanced algorithm then interprets the vector to achieve reconstruction of the spectrum. This scheme shifts the physical complexity of conventional grating- or interference-based spectrometers to computation. Moreover, many of the recent developments go well beyond optical spectroscopy, and we discuss them within a common framework, dubbed “geometric deep optical sensing.” The term “geometric” is intended to emphasize that in this sensing scheme, the physical properties of an unknown light beam and the corresponding photoresponses can be regarded as points in two respective high-dimensional vector spaces and that the sensing process can be considered to be a mapping from one vector space to the other. The mapping can be linear, nonlinear, or highly entangled; for the latter two cases, deep artificial neural networks represent a natural choice for the encoding and/or decoding processes, from which the term “deep” is derived. In addition to this classical geometric view, the quantum geometry of Bloch electrons in Hilbert space, such as Berry curvature and quantum metrics, is essential for the determination of the polarization-dependent photoresponses in some optical sensors. In this Review, we first present a general perspective of this sensing scheme from the viewpoint of information theory, in which the photoresponse measurement and the extraction of light properties are deemed as information-encoding and -decoding processes, respectively. We then discuss demonstrations in which a reconfigurable sensor (or an array thereof), enabled by device reconfigurability and the implementation of neural networks, can detect the power, polarization state, wavelength, and spatial features of an incident light beam. OUTLOOK As increasingly more computing resources become available, optical sensing is becoming more computational, with device reconfigurability playing a key role. On the one hand, advanced algorithms, including deep neural networks, will enable effective decoding of high-dimensional photoresponse vectors, which reduces the physical complexity of sensors. Therefore, it will be important to integrate memory cells near or within sensors to enable efficient processing and interpretation of a large amount of photoresponse data. On the other hand, analog computation based on neural networks can be performed with an array of reconfigurable devices, which enables direct multiplexing of sensing and computing functions. We anticipate that these two directions will become the engineering frontier of future deep sensing research. On the scientific frontier, exploring quantum geometric and topological properties of new quantum materials in both linear and nonlinear light-matter interactions will enrich the information-encoding pathways for deep optical sensing. In addition, deep sensing schemes will continue to benefit from the latest developments in machine learning. Future highly compact, multifunctional, reconfigurable, and intelligent sensors and imagers will find applications in medical imaging, environmental monitoring, infrared astronomy, and many other areas of our daily lives, especially in the mobile domain and the internet of things. Schematic of deep optical sensing. The n -dimensional unknown information ( w ) is encoded into an m -dimensional photoresponse vector ( x ) by a reconfigurable sensor (or an array thereof), from which w′ is reconstructed by a trained neural network ( n ′ = n and w′ ≈ w ). Alternatively, x may be directly deciphered to capture certain properties of w . Here, w , x , and w′ can be regarded as points in their respective high-dimensional vector spaces ℛ n , ℛ m , and ℛ n ′ .
Gong, Qixu; Cao, Huiping; Nagarkar, Parth(
, 2019 IEEE 35th International Conference on Data Engineering (ICDE))
Skyline queries are used to find the Pareto optimal solution from datasets containing multi-dimensional data points. In this paper, we propose a new type of skyline queries whose evaluation is constrained by a multi-cost transportation network (MCTN) and whose answers are off the network. This type of skyline queries is useful in many applications. For example, a person wants to find an apartment by considering not only the price and the surrounding area of the apartment, but also the transportation cost, time, and distance between the apartment and his/her work place. Most existing works that evaluate skyline queries on multi-cost networks (MCNs), which are either MCTNs or road networks, find interesting objects that locate on edges of the networks. Formally, our new type of skyline queries takes as input an MCTN, a query point q, and a set of objects of interest D with spatial information, where q and the objects in D are off the network. The answers to such queries are objects in D that are not dominated by other D objects when considering the multiple attributes of these objects and the multiple network cost from q to the solution objects. To evaluate such queries, we propose an exact search algorithm and its improved version by implementing several properties. The space of the exact skyline solutions is huge and can easily reach the order of thousands and incur long evaluation time. We further design much more efficient heuristic methods to find approximate solutions. We run extensive experiments using both real and synthetic datasets to test the effectiveness and efficiency of our proposed approaches. The results show that the exact search algorithm can be dramatically improved by utilizing several properties. The heuristic approaches to find approximate answers can largely reduce the query time and retrieve results that are comparable to the exact solutions.
Boffi, Nicholas M.; Vanden-Eijnden, Eric(
, Machine Learning: Science and Technology)
Abstract
The method of choice for integrating the time-dependent Fokker–Planck equation (FPE) in high-dimension is to generate samples from the solution via integration of the associated stochastic differential equation (SDE). Here, we study an alternative scheme based on integrating an ordinary differential equation that describes the flow of probability. Acting as a transport map, this equation deterministically pushes samples from the initial density onto samples from the solution at any later time. Unlike integration of the stochastic dynamics, the method has the advantage of giving direct access to quantities that are challenging to estimate from trajectories alone, such as the probability current, the density itself, and its entropy. The probability flow equation depends on the gradient of the logarithm of the solution (its ‘score’), and so isa-prioriunknown. To resolve this dependence, we model the score with a deep neural network that is learned on-the-fly by propagating a set of samples according to the instantaneous probability current. We show theoretically that the proposed approach controls the Kullback–Leibler (KL) divergence from the learned solution to the target, while learning on external samples from the SDE does not control either direction of the KL divergence. Empirically, we consider several high-dimensional FPEs from the physics of interacting particle systems. We find that the method accurately matches analytical solutions when they are available as well as moments computed via Monte-Carlo when they are not. Moreover, the method offers compelling predictions for the global entropy production rate that out-perform those obtained from learning on stochastic trajectories, and can effectively capture non-equilibrium steady-state probability currents over long time intervals.
The prevalence of emerging organic contaminants (EOCs) in ground and surface water has sparked the search for more effective methods to remove EOCs from the environment. In pursuit of a solution for this environmental concern, herein we present the development of reusable films based on cellulose nanofibers (CNFs) and the block copolymer, poly(4-vinylpyridine-b-ethylene oxide) (P4VP-PEO) to adsorb sulfamethoxazole (SMX) as an EOC model compound. We hypothesize that the adsorption of SMX was achieved mainly by π-π interactions between the pyridine functionalities of the block copolymer and the electron deficient phenyl group of the SMX. Preceding preparation of the films, CNFs were modified with the alkoxysilane trimethoxy(2-phenylethyl)silane (TMPES) to increase their stability in aqueous solution. After the addition of P4VP-PEO, the process was completed by filtration followed by oven-drying. XPS and FTIR were employed to confirm the addition of TMPES and P4VP-PEO, respectively. Adsorption batch experiments were performed in aqueous solutions of SMX at a neutral pH, obtaining adsorptions of up to 0.014 mmol/g in a moderate time of 60 min. For the reusability tests, films were immersed in ethanol 95 wt.% to elude the adsorbed SMX, rinsed with deionized (DI) water, and dried at room temperature to be reused in a new adsorption cycle. We found that this new composite material could be reused several times with negligible loss of adsorption capacity. The films presented have been shown to be of substantial importance for water remediation as they find direct application in the adsorption of electron deficient aromatic compounds and are reusable.
Kent, Spencer J., Frady, E. Paxon, Sommer, Friedrich T., and Olshausen, Bruno A. Resonator Networks, 2: Factorization Performance and Capacity Compared to Optimization-Based Methods. Retrieved from https://par.nsf.gov/biblio/10294578. Neural computation 32.12 Web. doi:10.1162/neco_a_01329.
Kent, Spencer J., Frady, E. Paxon, Sommer, Friedrich T., & Olshausen, Bruno A. Resonator Networks, 2: Factorization Performance and Capacity Compared to Optimization-Based Methods. Neural computation, 32 (12). Retrieved from https://par.nsf.gov/biblio/10294578. https://doi.org/10.1162/neco_a_01329
Kent, Spencer J., Frady, E. Paxon, Sommer, Friedrich T., and Olshausen, Bruno A.
"Resonator Networks, 2: Factorization Performance and Capacity Compared to Optimization-Based Methods". Neural computation 32 (12). Country unknown/Code not available. https://doi.org/10.1162/neco_a_01329.https://par.nsf.gov/biblio/10294578.
@article{osti_10294578,
place = {Country unknown/Code not available},
title = {Resonator Networks, 2: Factorization Performance and Capacity Compared to Optimization-Based Methods},
url = {https://par.nsf.gov/biblio/10294578},
DOI = {10.1162/neco_a_01329},
abstractNote = {We develop theoretical foundations of resonator networks, a new type of recurrent neural network introduced in Frady, Kent, Olshausen, and Sommer (2020), a companion article in this issue, to solve a high-dimensional vector factorization problem arising in Vector Symbolic Architectures. Given a composite vector formed by the Hadamard product between a discrete set of high-dimensional vectors, a resonator network can efficiently decompose the composite into these factors. We compare the performance of resonator networks against optimization-based methods, including Alternating Least Squares and several gradient-based algorithms, showing that resonator networks are superior in several important ways. This advantage is achieved by leveraging a combination of nonlinear dynamics and searching in superposition, by which estimates of the correct solution are formed from a weighted superposition of all possible solutions. While the alternative methods also search in superposition, the dynamics of resonator networks allow them to strike a more effective balance between exploring the solution space and exploiting local information to drive the network toward probable solutions. Resonator networks are not guaranteed to converge, but within a particular regime they almost always do. In exchange for relaxing the guarantee of global convergence, resonator networks are dramatically more effective at finding factorizations than all alternative approaches considered.},
journal = {Neural computation},
volume = {32},
number = {12},
author = {Kent, Spencer J. and Frady, E. Paxon and Sommer, Friedrich T. and Olshausen, Bruno A.},
editor = {null}
}
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