Re: Axiom-developer Digest, Vol 155, Issue 1

All of these research topics sound amazing to me. Are you submitting research proposals for them?

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> On Nov 7, 2016, at 10:00 AM, [email protected] wrote:
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> Today's Topics:
> 
>   1. Future research directions (Quantum Computing) (Tim Daly)
>   2. Future research directions (Universal Numbers) (Tim Daly)
>   3. Future research directions (Deep Neural Networks) (Tim Daly)
> 
> 
> ----------------------------------------------------------------------
> 
> Message: 1
> Date: Mon, 7 Nov 2016 04:13:20 -0500
> From: Tim Daly <[email protected]>
> To: axiom-dev <[email protected]>, Tim Daly
>    <[email protected]>
> Subject: [Axiom-developer] Future research directions (Quantum
>    Computing)
> Message-ID:
>    <CAJn5L=+GffD7RZc8Aks7M4XdYC+z=Q4SQdcDEAXsZH4mQie1zg@mail.gmail.com>
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> 
> Axiom is reaching a stable point where it can support research
> in new areas. There is current work on Gustafson's Universal
> Numbers (UNUMS) which promises to simplify the numeric
> libraries. It seems to also promise some symbolic/numeric
> computation. Time will tell.
> 
> Another area, still at the read-the-literature stage is quantum
> computing. IBM has made a 5-qubit machine available on the
> web. Some effort was devoted to implementing known algorithms.
> 
> Quantum computing had come a long way in the last few years.
> There are several attempts to create a high level language for
> expressing algorithms.
> 
> Quipper is an interesting effort:
> www.mathstat.dal.ca/~selinger/quipper
> It is currently implemented on top of Haskell but it looks like
> it could be implemented in Axiom. It uses Knill's QRAM model.
> Some papers:
> https://arxiv.org/pdf/1304.3390v1.pdf (Overview of the language)
> https://arxiv.org/pdf/1304.5485v1.pdf (Introductory examples)
> 
> My particular interest is twofold: Given my crypto background I'm
> interested in the post-crypto quantum work. I'm also interested in
> my observation that Hadamard gates are the basis for both unitary
> quantum operations and shared-channel telecommunications. By
> combining quantum crypto and shared-channel communication it
> seems that there is a potential for ultra-secure communication.
> 
> There are quantum computer simulators which could be embedded
> into Axiom to support simulated quantum development. Since the
> quantum operations are simple (unitary matrix operators) we could
> create a syntax and language front end to express them. For instance,
> being able to write a|Y> using the bra and ket notation.
> 
> This is a NIST-supported exploding area of research with many dozens
> of algorithms already available in the literature:
> http://math.nist.gov/quantum/zoo/
> 
> Tim
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> Message: 2
> Date: Mon, 7 Nov 2016 04:32:31 -0500
> From: Tim Daly <[email protected]>
> To: axiom-dev <[email protected]>, Tim Daly
>    <[email protected]>
> Subject: [Axiom-developer] Future research directions (Universal
>    Numbers)
> Message-ID:
>    <CAJn5L=J3EvGEHzFUfwUnG86HE0hwzgvR9_ty00+gAgzU=vQo7A@mail.gmail.com>
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> 
> Numeric computations have been locked into IEEE arithmetic
> for a long time. The problems are well known. NAG's numeric
> routines have a huge number of parameters to deal with these
> issues.
> 
> Axiom now has (growing) support for BLAS/LAPACK in native
> form. This gives us the opportunity to explore alternative
> number formats. BLAS routines could be adapted to use
> interval arithmetic but initial experiments show that this tends
> to generate wide, uninformative intervals.
> 
> Gustafson[0] has proposed a universal number format (UNUMs)
> which essentially allows two things. First, you can re-define the
> width of the exponent and mantissa fields to match the problem.
> Second, you can define exact intervals to generate exact results.
> 
> Efforts are underway to create a new UNUM number format which,
> hopefully, can be used in any context that DFLOAT can be used
> (e.g. in polynomials).
> 
> In the longer but still near term Intel has bought Altera so it now
> owns about 45% of the Field Programmable Gate Array (FPGA)
> market. They have used their manufacturing to shrink the die size
> by many generations and incorporated the FPGA onto the latest
> Intel processor chip. What this likely means is the ability to define
> new, novel instructions "on the fly" from your program. That would
> allow "hardware UNUM support".
> 
> Ultimately this has the potential to allow strong symbolic/numeric
> computation that does not suffer from overflow/underflow problems
> and allows "symbolic epsilons". These symbolic epsilons can
> eliminate the rounding-error / cancellation problems, leading to
> more exact arithmetic and dynamic error analysis.
> 
> Tim
> 
> [0] Gustafson, John "The End of Error" 13-978-1-4822-3986-7 (2015)
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> Message: 3
> Date: Mon, 7 Nov 2016 05:00:51 -0500
> From: Tim Daly <[email protected]>
> To: axiom-dev <[email protected]>, Tim Daly
>    <[email protected]>
> Subject: [Axiom-developer] Future research directions (Deep Neural
>    Networks)
> Message-ID:
>    <CAJn5L=LCcOUF2Cgx4Q=4uqGbwwPvot6Q5x59jvFasdWU9Q09VQ@mail.gmail.com>
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> 
> There is an explosion of research on Deep Neural Networks.
> OpenAI (https://openai.com/blog/) has 1 billion (US) dollars
> in support. They are doing focused work on reinforcement
> learning. Baidu, Google, Facebook, etc. are all deeply in the
> game.
> 
> Specifying Neural Networks to solve problems is easy. See
> https://www.youtube.com/watch?v=sEciSlAClL8 which explains
> the steps and the code to get 99+% accuracy on the MNIST
> dataset (handwritten digits).
> 
> Tensorflow includes primitives, such as 2d convolution,
> matrix multiplication, symbolic derivatives, RELU (rectified
> linear unit which is just zero for negative values and linear
> otherwise), sigmoid functions, atan functions, etc. The actual
> computation is just linear matrix computations, XW+B where
> X are the data, W are the weights, and B are the biases.
> 
> Latest "best practices" shows that deep neural networks are
> best implemented by repeating layers of XW+B followed by
> a non-linear (e.g. RELU, Sigmoid, Atan) step.
> 
> Axiom has the ability to do all of these tasks, making it a good
> platform for further research. In particular, there seems to be
> little "algorithmic analysis". The DNN area and NN research
> in general seems to be a collection of "tricks" (e.g. dropout).
> This is troubling since there is no easy way to predict the
> actual result, and rather frightening when the DNN is driving
> your car.
> 
> In theory what a DNN computes can be computed using a
> single layer NN. Can Axiom be used to "collapse" the layers
> by combining and spreading derivatives? A single layer NN
> with complicated derivatives seems easier to analyze than a
> multilayer iterated structure. The complicated derivatives could
> be "grouped" into similar classes and the shape of the higher
> order curves explored using symbolic expressions. This would
> give a clearer view of what the NN will do, where the high
> dimensional "valleys" lie, and where the system is sensitive.
> Such an ability to do analysis could reshape the industry.
> 
> Tim
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