
Deep Learning and Quantum Entanglement [pdf] - lainon
https://pdfs.semanticscholar.org/c931/6491b8d991fabbf9b28c449d66df6a50f841.pdf
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cs702
Here's my understanding after giving the paper a once-over: the authors
decompose a convnet with _linear_ activations (instead of, say, ReLUs, as is
common) and product-pooling layers (instead of, say, average- or max- pooling
layers, as is common) into tensors[1] that have the same structure as a
quantum many-body wave function. In other words, this convnet is equivalent to
such a function.

This equivalence allows the authors to use quantum entanglement measures to
quantify this convnet’s ability to model the correlation structures of its
inputs (i.e., its "expresiveness"), and tie that ability to the number of
feature maps in each convolutional layer.

I LOVE seeing these kinds of connections across different fields.

[1] By "tensor" I mean this:
[https://en.wikipedia.org/wiki/Tensor](https://en.wikipedia.org/wiki/Tensor)

~~~
DavidSJ
Is the quantum multiverse then the mind of Spinoza's God implemented as a
linear convent?

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mikhailfranco
My random stream-of-consciousness reference reactions ...

Tegmark & Lin's discussion of how deep learning maps onto the physical world:
_Why does deep and cheap learning work so well?_ [1][2]

The work of Aerts et al. more than 10 years ago, including how vector space
models of human categorization show QM structure [3].

Lucien Hardy's exquisite teasing out of the difference between classical and
quantum probability: _Quantum Theory From Five Reasonable Axioms_ [4]

It turns out that the innovation, power and strangeness of QM is related to
separating physical processes into:

1\. Linear continuous unitary reversible evolution relying on complex
amplitudes (wavefunction propagation).

2\. Non-linear discontinuous irreversible 'collapse' or 'measurement' or
'entanglement' or 'correlated branching', with probabilities based on the Born
Rule (real values from the square of wave function amplitudes).

Neural nets also decompose a problem into successive alternating layers of
reversible continuous functions and discrete irreversible categorical
decisions (e.g. softmax/sigmoid logistic classifiers).

An even more obscure tangent is that most of the financial market is now
constructed from 'options', which track continuous values with a continuous
payoff, but offer a choice of execution to collapse the option and create a
real financial result, e.g. see N.N.Taleb's dicussion of _optionality_ [5].

[1] [http://arxiv.org/pdf/1608.08225.pdf](http://arxiv.org/pdf/1608.08225.pdf)

[2]
[http://www.youtube.com/watch?v=5MdSE-N0bxs](http://www.youtube.com/watch?v=5MdSE-N0bxs)

[3]
[http://en.wikipedia.org/wiki/Diederik_Aerts#Quantum_structur...](http://en.wikipedia.org/wiki/Diederik_Aerts#Quantum_structure_in_psychology)

[4] [http://arxiv.org/abs/quant-ph/0101012](http://arxiv.org/abs/quant-
ph/0101012)

[5]
[http://fooledbyrandomness.com/ConvexityScience.pdf](http://fooledbyrandomness.com/ConvexityScience.pdf)

~~~
cs702
Thank you for posting this.

Similar thoughts have been percolating in my stream-of-consciousness for a
while, especially after coming across an earlier version of Tegmark & Lin's
paper some months ago.

My take is that deep neural nets work so well in practice because not all
distributions of natural data are equally likely (and therefore, the no-free-
lunch theorem(s), which assumes all distributions are equally likely, doesn't
hold in the real world!); and that, in turn, is because the distribution of
natural data is a consequence of the laws of Physics / symmetries of the
universe in which we happen to live.

PS. You will enjoy the following papers too: "An exact mapping between the
Variational Renormalization Group and Deep Learning" \-
[https://arxiv.org/abs/1410.3831](https://arxiv.org/abs/1410.3831) ; and
"Mutual Information, Neural Networks and the Renormalization Group" \-
[https://arxiv.org/abs/1704.06279](https://arxiv.org/abs/1704.06279) .

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Iv
Tl;dr: from what I understand, they use a mathematical tool used in physics to
measure the quality of some deep learning networks.

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bra-ket
the last co-author is the founder of Mobileye

