Experience & Education
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Google DeepMind
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Publications
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Prosthetic Devices as Goal-Seeking Agents
Peripheral-Nervous-System-Machine Interface Workshop at 2015 International Conference on Rehabilitation Robotics. Singapore.
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Magnetic Resonance Imaging of Blood Flow and Blood Oxygenation during Exercise
MSc Thesis, University of Alberta
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A new photoacoustic method for measuring optical transport green’s functions in turbid media
Paper presented at the Proceedings – IEEE Ultrasonics Symposium, 1246-1249.
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A photoacoustic method for optical scattering measurements in turbid media
Paper presented at the Progress in Biomedical Optics and Imaging – Proceedings of SPIE, 7177
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Combined photoacoustic and ultrasound biomicroscopy
Optics Express, 17(24), 22041-22046.
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Development of a fast-scanning combined ultrasound-photoacoustic biomicroscope
Paper presented at the Progress in Biomedical Optics and Imaging – Proceedings of SPIE, 7177.
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Photoacoustic technique for assessing optical scattering properties of turbid media
Journal of Biomedical Optics, 14(4), 040504.
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Ram Swaminathan
Recently, Steven Weng (a rising Sophomore at University of British Columbia and an intern at Slipbox.ai) asked me about the ability of a neural network to approximate any function arbitrarily close. His question got me to rethink about neural network approximation of continuous functions, and I wondered if there is a depth analogue to the width version of the universal approximation theorem for neural networks. Sure enough, there are some nice results in the literature. Let me explain. In the case of arbitrary width in the Universal Approximation Theorems, the number of neurons in the hidden layer is allowed to grow without bound while the depth remains fixed. It has been shown that a neural network with a single hidden layer and a suitable non-polynomial activation function such as sigmoid, ReLU, or tanh can approximate any continuous function on a compact domain to arbitrary precision. Increasing the number of neurons increases the number of folds or linear segments in the output, enabling increasingly accurate approximations. These results establish that shallow networks are universal approximators in theory, although the required width for complex functions can be very large. In the case of arbitrary depth, the width is fixed and the number of layers can grow. Research has shown that networks with width n+1 (“n” is the input dimension) and arbitrary depth can approximate any continuous function in a compact domain arbitrarily well. However, extremely narrow networks, such as those with width one using ReLU activation, have limited expressibility and cannot represent most functions. Increasing depth allows more complex piecewise-linear approximations, and with a minimum sufficient width, deep networks become universal approximators. What is the intuition behind why the width version seems to work differently from the depth version? A reasonable answer to this question, that I can think of, is as follows. A wide network with a single hidden layer and a large number of neurons essentially acts like a sum of simple functions, which are the activations of each neuron. Each neuron can be viewed as a basis function, similar to sine waves in Fourier series or polynomials in Taylor series. The network forms a linear combination of these basis functions after the nonlinear activation. By increasing the number of neurons, or the width, one can obtain a richer and more flexible basis set that can closely approximate or tile any target function’s shape on the input domain. In the depth version, deep networks reuse neurons across layers to build hierarchical or compositional features. Even when the depth is unlimited but the width is very small, the network has fewer basis functions to combine, so it may not approximate arbitrary functions as flexibly as a very wide network can. This is why some universality results for deep networks require a certain minimum width.
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Murad Hemmadi
New from me: All three Canadian firms in a key U.S. government program to prove out the groundbreaking potential of quantum computing have advanced to the next stage. Making it all the way through the Defense Advanced Research Projects Agency (DARPA) Quantum Benchmarking Initiative could open up to US$316 million in funding as well as lucrative contracts south of the border for Nord Quantique, Photonic Inc. and Xanadu. The Canadian firms' founders say reaching the program's second phase marks them as contenders in the disruptive technology field, and will help attract private-sector investment. The industry is also welcoming new funding in the federal budget, including $111.2 million designed to keep Canadian quantum companies in the country. But they're hopeful there's more to come from Ottawa, to stay competitive with the billions of dollars other nations have laid out. https://lnkd.in/gXT7dagU
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