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31b2f917e3f977c5587ded03c9b6cc69326be3af | subsection | 70 | 160 | Lower and upper matching bounds | Going back to (REF ):\liminf _{n_0 \rightarrow \infty } f_{\mathbf {n}}
&\ge {\sup }_{r_1 \in [0,]} {\inf }_{q_1 \in [0,\rho _1]} \underbrace{{\sup }_{q_0 \in [0,\rho _0]} {\inf }_{r_0 \ge 0} \; f_{\rm RS}\Big (q_0,r_0,q_1,r_1;\rho _0,\rho _1\Big )}_{= \psi (r_1, q_1)} \\
&= {\sup }_{r_1 \ge 0} {\inf }_{q_1 \in [0,\rho... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
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b77061841fc80d2ca2547eea6e5e37a548f03237 | subsection | 71 | 160 | Lower and upper matching bounds | Define for t \in [0,1]r_{\epsilon }(t) &R_1^{\prime }(t,\epsilon ) = 2\alpha _2 \Psi ^{\prime }_{P_{\rm out, 2}}\big (F_{\mathbf {n}}(t,y(t)),\rho _1(n_0)\big ) \in [0, r^*(n_0)] \subseteq [0,]\,;\\
q_{\epsilon }(t) &R_2^{\prime }(t,\epsilon ) = F_{\mathbf {n}}(t,R(t,\epsilon )) \in [0,\rho _1(n_0)]\,.Clearly R_1(t,\ep... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
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165eba2072300fa5b005ee89db4f745341b7adb6 | subsection | 72 | 160 | Lower and upper matching bounds | Therefore the Jacobian \det \big ({\partial R}{\partial \epsilon }\big ) is greater than or equal to 1. | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
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64557f93af8c94a0f34daa173b296470479a2cfd | subsection | 73 | 160 | Lower and upper matching bounds | We obtain that R is a \mathcal {C}^1-diffeomorphism (by the local inversion Theorem), and since its Jacobian is greater than or equal to 1 the functions (q_{\epsilon })_{\epsilon \in {\cal B}_{n_0}} and (r_{\epsilon })_{\epsilon \in {\cal B}_{n_0}} are regular.Proposition REF applied to this special choice of regular f... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
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0865487f33ceeb43c4e11bb74ff9e4c1826aa9ad | subsection | 74 | 160 | Lower and upper matching bounds | We chose the interpolation functions r_\epsilon and q_\epsilon such thatr_\epsilon (t) = 2 \alpha _2 \psi _{P_{\rm out, 2}}^{\prime }(q_\epsilon (t),\rho _1(n_0)) \,.Therefore q_\epsilon (t) is a critical point of the convex functionq_1 \in [0,\rho _1(n_0)] \mapsto \alpha \psi _{P_{\rm out, 2}}(q_1,\rho _1(n_0)) - \fra... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
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f39a8102cf8ff25981b628b9e9cf0a9a102bfbfe | subsection | 75 | 160 | Lower and upper matching bounds | Let f: [0,+\infty [ \rightarrow \mathbb {R} and g: [0,\rho _1] \rightarrow \mathbb {R} be the two functionsf(r_1) \!\!{\sup }_{q_0 \in [0,\rho _0]} {\inf }_{r_0 \ge 0} \Big \lbrace \psi _{P_0}(r_0) + \alpha _1 \Psi _{\varphi _1}(q_0, r_1;\rho _0) - \frac{r_0 q_0}{2} \Big \rbrace \,,\;
g(q_1) \alpha \Psi _{P_{\rm out,2}... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
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1a979fdb56511a6b86cfdbd10cc6324416073126 | subsection | 76 | 160 | Activations comparison in terms of mutual informations | Here we assume the exact same setting as the one presented in the main text to compare activation functions on a two-layer random weights network. We compare here the mutual information estimated with the proposed replica formula instead of the entropy behaviors discussed in the main text. As it was the case for entrop... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
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d2fdc6287a734df5e8277e670c6836445ba0f753 | subsection | 77 | 160 | Learning ability of USV-layers | To ensure weight matrices remain close enough to being independent during learning we introduce USV-layers, corresponding to a custom type of weight constraint. We recall that in such layers, weight matrices are decomposed in the manner of a singular value decomposition, W_{\ell } = U_{\ell }S_{\ell }V_{\ell }, with U_... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
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f6efd704acbedf00fed8fe965fbb556e988d72fd | subsection | 78 | 160 | Learning ability of USV-layers | All experiments use the same learning rate 0.01 and batchsize of 100 samples. Results are averaged over 5 independent runs, and standard deviations are reported in parentheses.] | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
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... |
474b5425648d1d68ad4f3ef9ae6a4d456e5ceb02 | subsection | 79 | 160 | Additional learning experiments on synthetic data | Similarly to the experiments of the main text, we consider simple training
schemes with constant learning rates, no momentum, and no explicit
regularization.We first include a second version of Figure 4 of the main text, corresponding to the exact same experiment with a different random seed and check that results are ... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
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0.02... |
2951fb25b42befba8ad35076d9f0a9ec0ed63270 | subsection | 80 | 160 | Additional learning experiments on synthetic data | Remaining: mutual information from each layer displayed separately.]In a last experiment, we even show that merely changing the weight
initialization can drastically change the behavior of mutual informations
during training while resulting in identical training and testing final
performances. We consider here a settin... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
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9ac8a6ce2e4aa1997e0b1025e563ab17943d8541 | subsection | 81 | 160 | Additional learning experiments on synthetic data | Learning and hidden-layers mutual information curves for a classification problem with correlated input data, using a 4-USV hardtanh layers and 1 unconstrained softmax layer, from 3 different initializations. Top: Initial weights at layer \ell of variance 4 / n_{\ell -1}, best training accuracy 0.999, best test accurac... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
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bd2b0d97a7fb01408eb982e70373371db5b0cff3 | subsection | 82 | 160 | The Nishimori identity | [Nishimori identity]
Let (,) \in ^{n_1} \times ^{n_2} be a couple of random variables. Let k \ge 1 and let ^{(1)}, \dots , ^{(k)} be k i.i.d. samples (given ) from the conditional distribution P(=\cdot \, | ), independently of every other random variables. Let us denote \langle - \rangle the expectation operator w.r.t... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
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9df25b91fe846e70173101f3433f494d69ff661f | subsection | 83 | 160 | Limit of the sequence | Here we prove Proposition REF , i.e. that the sequence (\rho _1(n_0))_{n_0 \ge 1} converges to \rho _1 [\varphi _1^2(T, _1)], where T \sim (0,\rho _0) and _1 \sim P_{A_1} are independent,
under the hypotheses REF REF REF .If \rho _0 = 0 then ^0 = 0 almost surely (a.s.) and \rho _1(n_0) = \varphi _1^2(0, _1) = \rho _1 f... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
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4689fb8b16d067b20401d6488ec3ba4c029ff69c | subsection | 84 | 160 | Limit of the sequence | Combined with the continuity of h, one has\lim _{n_0 \rightarrow +\infty } h\left(\frac{\Vert ^0 \Vert ^2}{n_0}\right)\,\stackrel{{\normalfont \mbox{a.s.}}}{=} \, h(\rho _0) = \rho _1 \:.Noticing that \left|h\left({\Vert ^0 \Vert ^2}{n_0}\right) \right|\le \sup \varphi _1^2, the dominated convergence theorem gives\rho ... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
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2712d7e6af428b82a39750e9526ab5543512b90a | subsection | 85 | 160 | Properties of the third scalar channel | Assume \varphi _1 is bounded (as it is the case under REF ). Let V,U (0,1) and \rho _0 \ge 0, q_0 \in [0,\rho _0].
For any r \ge 0, Y_0^{\prime (r)} = \sqrt{r}\varphi _1(\sqrt{q}\, V + \sqrt{\rho - q} \,U, _1) + Z^{\prime } where Z^{\prime } \sim (0,1), _1 \sim P_{A_1}. The function\Psi _{\varphi _1}(q_0, \, \cdot \, ;... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
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0... |
6e461102b8e1d7c84b6cc6169ca9aec57d2bcc41 | subsection | 86 | 160 | Properties of the third scalar channel | Denote \langle - \rangle _r the expectation operator w.r.t. the joint posterior distributiondP(u, \vert Y_0^{\prime }, V) = \frac{1}{(Y_0^{\prime }, V)}{\cal D}u \, dP_{A_1}() e^{\sqrt{r}y_0^{\prime } \varphi _1(\sqrt{q_0}V + \sqrt{\rho _0 - q_0} u,)-\frac{r}{2}\varphi _1^2(\sqrt{q_0}V + \sqrt{\rho _0 - q_0}u,)} \,,whe... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
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0.0... |
c867e8b88941acf850d83c3f4731d412d89366e3 | subsection | 87 | 160 | Derivative of the averaged interpolating free entropy | This appendix is dedicated to the proof of Proposition REF , i.e. to the derivation of the interpolating free entropy f_{\mathbf {n},\epsilon }(t) with respect to the time t.
First we show that for all t \in (0,1)\frac{df_{\mathbf {n},\epsilon }(t)}{dt} =
-\frac{1}{2} \frac{n_1}{n_0}\Bigg [\Bigg \langle \Bigg (\frac{1}... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.05453534796833992,
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... |
e8ff6467c4e6f9c90749ba8883a20f49270d7034 | subsection | 88 | 160 | Computing the derivative: proof of( | Recall definition (REF ). Once written as a function of the interpolating Hamiltonian (REF ), it becomesf_{\mathbf {n},\epsilon }(t) \!= \!\frac{1}{n_0} _{{1},{2},}\bigg [\int \!\! dd^{\prime } dP_0(^0)dP_{A_1}(_1) {\cal D}(2\pi )^{-\frac{n_1}{2}} e^{-_{t,\epsilon }(^0,_1,;,^{\prime },{1},{2},)}\\
\cdot \ln \int dP_0()... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
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0.0... |
3f08968786b67b4f1180a835229358278175b9af | subsection | 89 | 160 | Computing the derivative: proof of( | \mathbf {n},t,\epsilon } \,\Big ] }_{T_2}\,,where _{\mathbf {n},t,\epsilon } \equiv _{\mathbf {n},t,\epsilon }(,^{\prime },{1}, {2},) is defined in (REF ).
In the remaining part of this subsection REF , to lighten notations,
the second argument of the function \varphi _1 will be omitted (except in a few occasions). Nam... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
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0.0... |
94cc8f817fbb252215e8ae9228f48aaba67db388 | subsection | 90 | 160 | Computing the derivative: proof of( | For 1 \le \mu \le n_2 one has from (REF )-\bigg [\frac{dS_{t,\epsilon ,\mu }}{dt} u^{\prime }_{Y_\mu }(S_{t,\epsilon ,\mu }) \ln _{\mathbf {n},t,\epsilon } \bigg ]
=\frac{1}{2}\bigg [
\frac{1}{\sqrt{n_1 (1-t)}}\bigg [{2} \varphi _1\bigg (\frac{{1} ^0}{\sqrt{n_0}}\bigg )\bigg ]_{\mu } u^{\prime }_{Y_\mu }(S_{t,\epsilon ... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.020824480801820755,
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0.0... |
95ccdf7f0287faea42aa03d40f25de0ee5711df1 | subsection | 91 | 160 | Computing the derivative: proof of( | \mathbf {n},t,\epsilon }\,\Bigg ]\\
&\quad =\Bigg [\frac{\big \Vert ^1\big \Vert ^2}{n_1}
\frac{P_{\rm out,2}^{\prime \prime }(Y_{\mu } | S_{t,\epsilon ,\mu })}{P_{\rm out,2}(Y_{\mu } | S_{t,\epsilon ,\mu })}
\ln _{\mathbf {n},t,\epsilon } \Bigg ]
+ \Big [\big \langle \widehat{Q} \:
u_{Y_{\mu }}^{\prime } ( S_{t,\epsil... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
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0.022315530106425285,
-0.027352552860975266,... |
3d321f907c7e952178d2104c19f8464d35d245e8 | subsection | 92 | 160 | Computing the derivative: proof of( | Using again Gaussian integration by parts, but this time w.r.t V_\mu , U_\mu {\cal N}(0,1), one similarly obtains&\Bigg [
\Bigg ( \frac{q_{\epsilon }(t)}{ \sqrt{R_2(t,\epsilon )}} V_{\mu }
+ \frac{\rho _1(n_0) - q_{\epsilon }(t)}{\sqrt{\rho _1(n_0)t + 2s_n - R_2(t,\epsilon )}} U_{\mu }\Bigg )
u_{Y_{\mu }}^{\prime } ( S... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.0019386401399970055,
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0.035610709339380264,
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0.010947166010737419,
0.006980248726904392,... |
efc6e6193591e3dce7022b9cb54187699b30c6bb | subsection | 93 | 160 | Computing the derivative: proof of( | \mathbf {n},t,\epsilon }\,.Combining equations (REF ), (REF ) and (REF ) together gives us-\bigg [\frac{dS_{t,\epsilon ,\mu }}{dt} u^{\prime }_{Y_\mu }(S_{t,\epsilon ,\mu }) \ln _{\mathbf {n},t,\epsilon } \bigg ]
= \frac{1}{2}\Bigg [
\frac{P_{\rm out,2}^{\prime \prime }(Y_\mu | S_{t,\epsilon ,\mu })}{P_{\rm out,2}(Y_\m... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.02922075428068638,
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0.035370081663131714,
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0.02... |
97c17397687f8951b57160a72f12daf0080d0a87 | subsection | 94 | 160 | Computing the derivative: proof of( | It comes&\bigg [\varphi _1\bigg ( \bigg [\frac{{1} ^0}{\sqrt{n_0}}\bigg ]_{i}\bigg )
\bigg (Y^{\prime }_i - \sqrt{R_1(t,\epsilon )} \varphi _1\bigg ( \bigg [\frac{{1} ^0}{\sqrt{n_0}}\bigg ]_{i}\bigg )\bigg )
\ln _{\mathbf {n},t,\epsilon } \bigg ]&&\qquad \qquad =\bigg [\varphi _1\bigg ( \bigg [\frac{{1} ^0}{\sqrt{n_0}}... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
0.00418039271607995,
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-0.022656509652733803,
-... |
04457e2b21b51dd01ebf943b9b8c4809980d7931 | subsection | 95 | 160 | Computing the derivative: proof of( | \mathbf {n},t,\epsilon } \,\bigg ) \,.After taking the sum over i \in \lbrace 1,\dots ,n_1\rbrace , we get-\frac{1}{2}\frac{r_{\epsilon }(t)}{\sqrt{R_1(t,\epsilon )}}
\Bigg [ \frac{1}{n_0} \sum _{i=1}^{n_1}
\varphi _1\bigg ( \bigg [\frac{{1} ^0}{\sqrt{n_0}}\bigg ]_{i}\bigg )
\bigg (Y^{\prime }_i - \sqrt{R_1(t,\epsilon ... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.01980462856590748,
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0.007583128288388252,
-0.012663976289331913,
... |
27b30867fb18947a76b97daf091012c8774d2396 | subsection | 96 | 160 | Computing the derivative: proof of( | The Nishimori identity (see Proposition REF ) saysT_2 = \frac{1}{n_0} \Big [\big \langle _{t,\epsilon }^{\prime }(,_1,;,^{\prime },{1},{2},) \big \rangle _{\! \mathbf {n},t,\epsilon }\Big ]\\
= \frac{1}{n_0} \big [_{t,\epsilon }^{\prime }(^0,_1,;,^{\prime },{1},{2},)\big ].From (REF ) it directly comes&\big [_{t,\epsil... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.014667239971458912,
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-0.0176800116896629... |
ffc64172fdb63b9c2706180ff4ec147d5c4f90e8 | subsection | 97 | 160 | Proof that | The last step to prove Proposition REF is to show that A_{\mathbf {n},\epsilon }(t) – see definition (REF ) – vanishes uniformly in {t \in [0,1]} and \epsilon as n_0 \rightarrow +\infty , under conditions REF -REF -REF . First we show thatf_{\mathbf {n},\epsilon }(t) \cdot \Bigg [
\sum _{\mu =1}^{n_2} \frac{P_{\rm out,... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.01552931871265173,
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0.018015841022133827,
-0.009091821499168873... |
16d9165f53b8666be039664fbd1f194e83a947c6 | subsection | 98 | 160 | Proof that | Consequently, for {\mu \in \lbrace 1 ,\dots , n_2 \rbrace },\bigg [
\frac{P_{\rm out,2}^{\prime \prime }(Y_{t,\epsilon ,\mu } | S_{t,\epsilon ,\mu })}{P_{\rm out,2}(Y_{t,\epsilon ,\mu } | S_{t,\epsilon ,\mu })} \, \bigg | \, ^1, _{t,\epsilon } \bigg ]
= \int dY_\mu P_{\rm out,2}^{\prime \prime }(Y_\mu |S_{t,\epsilon ,\... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.03146642446517944,
0.05740867182612419,
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0.005012958310544491,
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0.01267355214804411,
-0.02745300531387329,
0.... |
614b3633f556a140713871d5667170fe27afb6ac | subsection | 99 | 160 | Proof that | Using successively (REF ) and the Cauchy-Schwarz inequality, we have\big \vert A_{\mathbf {n},\epsilon }(t) \big \vert &= \Bigg \vert \Bigg [
\sum _{\mu =1}^{n_2} \frac{P_{\rm out,2}^{\prime \prime }(Y_{t,\epsilon ,\mu } | S_{t,\epsilon ,\mu })}{P_{\rm out,2}(Y_{t,\epsilon ,\mu } | S_{t,\epsilon ,\mu })}
\Bigg (\frac{\... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.03909722715616226,
0.04858921840786934,
0.008713707327842712,
-0.017244288697838783,
0.006169029977172613,
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0.021547731012105942,
0.019228145480155945,
0.009057066403329372,
0.004948195070028305,
-0.04770411178469658,
0.032687846571207047,
-0.029513677582144737,
0.02... |
267dbbd2dff9007f9c57fd73476c0bdab9a808d4 | subsection | 100 | 160 | Proof that | 2}\,
\Bigg ]\\
= \Bigg [
\Bigg [
\Bigg (\sum _{\mu =1}^{n_2} \frac{P_{\rm out,2}^{\prime \prime }(Y_{t,\epsilon ,\mu } | S_{t,\epsilon ,\mu })}{P_{\rm out,2}(Y_{t,\epsilon ,\mu } | S_{t,\epsilon ,\mu })}\Bigg )^{\!\! 2}
\, \Bigg | \, ^1, _{t,\epsilon }
\Bigg ] \cdot \Bigg (\frac{\big \Vert ^1\big \Vert ^2}{n_1} - \rho ... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.026000360026955605,
0.030181169509887695,
-0.02973867394030094,
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-0.022246787324547768,
0.024062540382146835,
-0.02046154998242855,
... |
a25bd85c11b84a4ee955a95940bcdc1ab53f252d | subsection | 101 | 160 | Proof that | 2}
\, \Bigg | \, _{t,\epsilon }
\Bigg ].Under condition REF , it is not difficult to show that there exists a constant C > 0 such that\bigg [ \bigg (\frac{P_{\rm out,2}^{\prime \prime }(Y_{t,\epsilon ,1} | S_{t,\epsilon ,1})}{P_{\rm out,2}(Y_{t,\epsilon ,1} | S_{t,\epsilon ,1})}\bigg )^2 \, \bigg | \, _{t,\epsilon } \b... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.0475984551012516,
0.048025622963905334,
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-0.008078007027506828,
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0.035942938178777695,
0.04976479709148407,
0.010839328169822693,
-0.039024632424116135,
-0.017300209030508995,
-0.004996312316507101,... |
a581a288359fe626708fb7d7e47f3ec4e0909a22 | subsection | 102 | 160 | Proof that | Conditionally on ^0, the random variables (X_i^1)_{1 \le i \le n_1} are i.i.d. and{\mathbb {V}\mathrm {ar}}\big (\big \Vert ^1 \big \Vert ^2 \big \vert ^0 \big )
= \sum _{i=1}^{n_1} {\mathbb {V}\mathrm {ar}}\big (\big (X_i^1\big )^2 \big \vert ^0 \big )
= n_1 \, {\mathbb {V}\mathrm {ar}}\big (\big (X_1^1\big )^2 \big \... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.05180611461400986,
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... |
bd19f22b6515cfc951d0d22c909fdb26b5bab63f | subsection | 103 | 160 | Proof that | The partial derivatives of g satisfy for 1 \le j \le n_0\frac{\partial g}{\partial c_j}()
&= \bigg [2\varphi _1\bigg ( \bigg [\frac{{1} }{\sqrt{n_0}}\bigg ]_{1}, _{1,1} \bigg )\varphi ^{\prime }_1\bigg ( \bigg [\frac{{1} }{\sqrt{n_0}}\bigg ]_{1}, _{1,1} \bigg ) \frac{({1})_{1j}}{\sqrt{n_0}}\bigg ]&= \frac{2 c_j}{n_0}\b... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
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-0.01568978652358055... |
be9ee3c1e09705343b4551635483970e1f523135 | subsection | 104 | 160 | Proof that | \forall j \in \lbrace 1,\dots ,n_0\rbrace\sup _{\in \mathcal {X}^{n_0}, c^{\prime }_j\in \mathcal {X}} \big \vert g() - g(c_1, \dots ,c^{\prime }_j,\dots , c_{n_0}) \big \vert \le \frac{C}{n_0} \sup _{c_j, c^{\prime }_j\in \mathcal {X}} \big \vert c_j - c^{\prime }_j \big \vert \le \frac{2S \cdot C}{n_0}\, .Applying Pr... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
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0.008787672966718674,... |
7fba2809183351b680bc833d3bfeda061dc1a1a9 | subsection | 105 | 160 | Concentration of the free entropy | In this section, we prove that the free entropy of the interpolation model studied in Sec. REF concentrates around its expectation (uniformly in t and \epsilon ), i.e. we prove Theorem REF stated below.
C(\varphi _1, \varphi _2, \alpha _1, \alpha _2, S) will denote any generic positive constant depending only on \varph... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.028579136356711388,
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0.0... |
5674ecb4c937e92eb31a1ceebd2bc23f1b870233 | subsection | 106 | 160 | Concentration of the free entropy | The interpolating Hamiltonian (REF )-(REF ) is- \sum _{\mu =1}^{n_2}
\ln P_{\rm out,2} ( Y_{\mu } |s_{t,\epsilon }(, _1, u_\mu )) + \frac{1}{2} \sum _{i=1}^{n_1}\bigg (Y^{\prime }_{i} - \sqrt{R_1(t,\epsilon )}\, \varphi _1\bigg (\bigg [\frac{{1} }{\sqrt{n_0}}\bigg ]_i, _{1,i}\bigg )\bigg )^2 \; ,where s_{t, \epsilon , ... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
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0.... |
dd95708f45c437847ceb8a8e68374dac04b55fe9 | subsection | 107 | 160 | Concentration of the free entropy | To lighten notations, one also defines ^1 \equiv ^1(,_1) \varphi _1\Big (\frac{{1} }{\sqrt{n_0}}, _{1}\Big ). The channel P_{\rm out,2} defined in (REF ) can be written asP_{\rm out,2}(Y_{t,\epsilon ,\mu } | s_{t, \epsilon , \mu }(, _1, ))
&= \! \int \! dP_{A_2}(_{2,\mu }) \frac{1}{\sqrt{2\pi \Delta }}
e^{-\frac{1}{2\D... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
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... |
8343c93280376a9aeaaaaa1c2a4dc4402e1e4d68 | subsection | 108 | 160 | Concentration of the free entropy | \int \! dP_{A_2}(_{2,\mu }) \frac{1}{\sqrt{2\pi \Delta }}e^{-\frac{1}{2\Delta }(\Gamma _{t,\epsilon ,\mu }(, _1, _{2,\mu }, u_\mu )+ \sqrt{\Delta } Z_{\mu })^2 }with\Gamma _{t,\epsilon ,\mu }(, _1, _{2,\mu }, u_\mu )
\varphi _2\bigg (
\sqrt{\frac{1-t}{n_1}}\, \big [{2} ^1 \big ]_\mu + k_1(t) \,V_{\mu } + k_2(t) \,U_{\m... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
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0.011113906279206276,
0.013440428301692009... |
2a8425bdbacc5d3d5438d5ac79f85e815cd3edec | subsection | 109 | 160 | Concentration of the free entropy | Our goal is to show that the free energy (REF ) concentrates around its expectation.
We will prove that there exists a positive constant C(\varphi _1, \varphi _2, \alpha _1, \alpha _2, S) such that the variance of {\ln \hat{}_{\mathbf {n},t,\epsilon }}{n_0} is bounded by {C(\varphi _1, \varphi _2, \alpha _1, \alpha _2,... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
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0.017731640487909317,
... |
a12eea7c9c95462041b0e3381d2251800330ba51 | subsection | 110 | 160 | Concentration of the free entropy | Then the concentration w.r.t. {1} and ^1 \varphi _1\big ({{1}^0}{\sqrt{n_0}}, _1\big ) is obtained by proving the concentation w.r.t. _1, {1} and ^0, in this order.Before starting the proof of Theorem REF we point out that, under REF , all the suprema \sup \vert \varphi _k \vert , \sup \vert \varphi _k^{^{\prime }} \ve... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
0.014800749719142914,
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0.0030421644914895296,
... |
e21be1804d48ecfaac7ff701bfdd3d66366adc96 | subsection | 111 | 160 | Concentration conditionally on | In all this subsection, we work conditionally on {1}, ^1 and we prove that {\ln \widehat{}_{\mathbf {n},t,\epsilon }}{n_0} is close to its expectation w.r.t. all the other random variables, namely , ^{\prime }, , , {2} and _2:
Under REF REF REF , there exists a positive constant {C(\varphi _1, \varphi _2, \alpha _1... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.029320038855075836,
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0.014308972284197807,
0.0032777085434645414,
... |
9732f2dea58df0fa0222aa755a4c4564720e8a75 | subsection | 112 | 160 | Concentration conditionally on | One finds\left|\frac{\partial g}{\partial Z_\mu }\right|&= \frac{1}{n_0\sqrt{\Delta }} \big \vert \langle \Gamma _{t,\mu } \rangle _{\widehat{\mathcal {H}}_{t,\epsilon }}\big \vert \le \frac{2}{n_0\sqrt{\Delta }} \sup \vert \varphi _2 \vert \;,\\
\left|\frac{\partial g}{\partial Z_i^\prime }\right|&= \frac{1}{n_0} \Big... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.033773552626371384,
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0.01851901225745678,
-0.028922609984874725,... |
1ff2c64ab9cc9023fcb6cf263c69076e2d3f0875 | subsection | 113 | 160 | Concentration conditionally on | Under REF REF REF , there exists a positive constant {C(\varphi _1, \varphi _2, \alpha _1, \alpha _2, S)} such that \forall t \in [0,1]\Bigg [\Bigg (
\bigg [\frac{\ln \widehat{}_{\mathbf {n},t,\epsilon }}{n_0} \bigg \vert , , {2}, _2, ^1, {1}\bigg ]
- \bigg [\frac{\ln \widehat{}_{\mathbf {n},t,\epsilon }}{n_0} \bigg ... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.04319073632359505,
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0.006406117230653763,
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-0.010828208178281784,
-0.005948265083134174,... |
e35de25a2a53923653b0f1c94c790dea998e5cc5 | subsection | 114 | 160 | Concentration conditionally on | ({2})_{\mu i}, first remark that\frac{\partial \Gamma _{t, \epsilon , \mu }}{\partial ({2})_{\mu i}}
= \sqrt{\frac{1-t}{n_1}} \bigg (
X_i^1\varphi _2^{^{\prime }}\bigg (
\sqrt{\frac{1-t}{n_1}}\, \big [{2} ^1\big ]_\mu + k_1(t) \,V_{\mu } + k_2(t) \,U_{\mu }\bigg )\\
- x_i^1 \varphi _2^{^{\prime }}\bigg (\sqrt{\frac{1-t... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.0491940900683403,
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-0.0037307552993297577,
-0.0490720197558403,
... |
2869f0f4909eaddecd6a2875c0d6f798f386061c | subsection | 115 | 160 | Concentration conditionally on | \widehat{\mathcal {H}}_{t,\epsilon }}\bigg ]\bigg \vert \\
&\le \frac{1}{n_0 \sqrt{n_1}}\tilde{}\Big [(2\sup \vert \varphi _2\vert + \sqrt{\Delta } \vert Z_\mu \vert ) \Delta ^{-1}
(2 \sup \vert \varphi _1 \vert \sup \vert \varphi _2^{^{\prime }} \vert )\Big ] \\
&= \frac{1}{n_0 \sqrt{n_1}}\bigg (2\sup \vert \varphi _2... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.018458962440490723,
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-0.010053269565105438,
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0.025857802480459213,
-0.010076153092086315,
0.... |
6593dfabff95f8a8dd9f6c705b734f96226d18b4 | subsection | 116 | 160 | Concentration conditionally on | Under REF REF REF , there exists a positive constant {C(\varphi _1, \varphi _2, \alpha _1, \alpha _2, S)} such that \forall t \in [0,1]\Bigg [\Bigg (
\bigg [\frac{\ln \widehat{}_{\mathbf {n},t,\epsilon }}{n_0} \bigg \vert _2, ^1, {1} \bigg ]
- \bigg [\frac{\ln \widehat{}_{\mathbf {n},t,\epsilon }}{n_0} \bigg \vert ^1... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.023328278213739395,
0.026913722977042198,
-0.04024852439761162,
-0.017149105668067932,
-0.005290438421070576,
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0.01666087657213211,
0.004706849809736013,
-0.012602456845343113,
-0.011183536611497402,
0.0036502880975604057,
... |
658a693b59982af16d1efee008cda55f05705b45 | subsection | 117 | 160 | Concentration conditionally on | By an application of Jensen's inequality one finds\frac{1}{n_0} \mathbb {E}_{G}\Big [ \big \langle \widehat{\mathcal {H}}_{t,\epsilon }^{(\nu )} - \widehat{\mathcal {H}}_{t,\epsilon } \big \rangle _{\widehat{\mathcal {H}}_{t,\epsilon }^{(\nu )}} \Big ]
\le g(_2) - g(_2^{(\nu )})
\le \frac{1}{n_0} \mathbb {E}_{G}\Big [ ... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.024136722087860107,
0.057214103639125824,
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0.01632509008049965,
0.0005926427547819912,
-0.04095004126429558,
0.013449128717184067,
-0.002660455647855997,
... |
4df6579d4f22bcf53873b54269eac13ac431a104 | subsection | 118 | 160 | Concentration conditionally on | From (REF ) we conclude that g satisfies the bounded difference property:\Big \vert g\big (_2\big ) - g\big (_2^{(\nu )}\big ) \Big \vert \le \frac{2\sup \vert \varphi _2\vert }{\Delta n_0} \Bigg ( 2 \sup \vert \varphi _2\vert + \sqrt{\frac{2}{\pi }\Delta }\Bigg ).An application of Proposition REF (remember _{2,1}, \do... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.04172937944531441,
0.024115735664963722,
-0.008860243484377861,
-0.02712257206439972,
-0.019124694168567657,
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-0.008684717118740082,
0.017262592911720276,
0.014484704472124577,
-0.006799721624702215,
-0.030053090304136276,
0.017354171723127365,
-0.03699781373143196,
... |
7f6da36862761706762f27a7cdeb8a205fc00581 | subsection | 119 | 160 | Concentration with respect to | In this subsection, we prove that {[\ln \widehat{}_{\mathbf {n},t,\epsilon } \vert X^1 , {1}]}{n_0} is close to its expectation, i.e.
Under REF REF REF , there exists a positive constant {C(\varphi _1, \varphi _2, \alpha _1, \alpha _2, S)} such that \forall t \in [0,1]\Bigg [\Bigg (
\bigg [\frac{\ln \widehat{}_{\mat... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.023131197318434715,
0.04925296828150749,
-0.05743128061294556,
-0.023695744574069977,
-0.005782799329608679,
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-0.0056645493023097515,
0.010009277611970901,
0.01337367668747902,
0.008491101674735546,
-0.016677042469382286,
-0.006477040238678455,
0.01047464832663536,
... |
2d25c40dfe98fff4dd00005118b795a7f43f6c92 | subsection | 120 | 160 | Concentration with respect to | Define {g() = [{\ln \widehat{\mathcal {Z}}_{\mathbf {n},t,\epsilon }}{n_0} \vert ^1 = , {1},^0]}. We will show that g satisfies the bounded difference property, then an application of Proposition REF will end the proof.Let i \in \lbrace 1, \dots , n_1\rbrace . Consider two vectors {,^{(i)} \in [-\sup \vert \varphi _1 \... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.02215363271534443,
0.013571389019489288,
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0.005858812015503645,
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0.01734757609665394,
0.02422862872481346,
-0.03420691937208176,
0.006507248152047396,
-0.016477908939123154,
-0.... |
6a3d159f93add715552f76afbef50ca224f45e25 | subsection | 121 | 160 | Concentration with respect to | \hat{}_{t,\epsilon }}\,\bigg ]&\le \,\frac{2 \sup \vert \varphi _1 \vert }{n_0 \Delta }
\sum _{\mu =1}^{n_2} \bigg \vert \tilde{}\bigg [\bigg \langle (\Gamma _{t,\epsilon ,\mu } + \sqrt{\Delta }Z_\mu ) \frac{\partial \Gamma _{t,\epsilon ,\mu }}{\partial X_i^1} \bigg \rangle _{\!\widehat{\mathcal {H}}_{t,\epsilon }} \bi... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.012981103733181953,
0.03733307495713234,
-0.027229027822613716,
-0.006040296051651239,
-0.022955413907766342,
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0.005010049790143967,
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0.025733264163136482,
0.013812932185828686,
-0.03748570382595062,
0.008158024400472641,
-0.00005407600838225335,... |
f1bae3e432d242a0b35b3b16637f76401f4a910e | subsection | 122 | 160 | Concentration with respect to | ({2})_{\mu i} in the following expectation:&\bigg \vert \tilde{}\bigg [\bigg \langle \!(\Gamma _{t,\epsilon ,\mu } + \sqrt{\Delta }Z_\mu ) \frac{\partial \Gamma _{t,\epsilon ,\mu }}{\partial X_i^1} \bigg \rangle _{\!\widehat{\mathcal {H}}_{t,\epsilon }}\,\bigg ]\bigg \vert \\
&\;= \bigg \vert \tilde{}\bigg [\bigg \lang... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.01147602591663599,
0.029269970953464508,
-0.0510011687874794,
-0.01883167028427124,
-0.021639633923768997,
-0.008095786906778812,
0.01472654938697815,
0.0005145707982592285,
0.01845015399158001,
0.0037312344647943974,
-0.028583239763975143,
-0.004879600368440151,
0.006977942772209644,
0... |
7c11be7be0fe3f6243175f8730b60d0616819ad2 | subsection | 123 | 160 | Concentration with respect to | The condition (REF ) is thus satisfied and Proposition REF implies that\Bigg [\Bigg (
\bigg [\frac{\ln \hat{}_{\mathbf {n},t,\epsilon }}{n_0} \bigg \vert ^1,{1}, ^0 \bigg ] - \bigg [\frac{\ln \hat{}_{\mathbf {n},t,\epsilon }}{n_0} \bigg \vert {1}, ^0\bigg ]
\Bigg )^2 \Bigg \vert {1}, ^0 \Bigg ]
\le \frac{C(\varphi _1, ... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.03545532748103142,
0.04677540063858032,
-0.024440376088023186,
-0.03197691962122917,
-0.006132978014647961,
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0.013105045072734356,
-0.01254056766629219,
-0.007738689426332712,
-0.02026018686592579,
0.0... |
d32fda06400680bf8b086dfd2ea44d73871ca11d | subsection | 124 | 160 | Concentration with respect to | \widehat{\mathcal {H}}_{t,\epsilon }} \, \bigg ]\\
-\frac{\sqrt{R_1(t,\epsilon )}}{n_0^{3/2}} \mathbb {E}\bigg [\bigg \langle \bigg (\sqrt{R_1(t,\epsilon )} X_i^1 - \sqrt{R_1(t,\epsilon )} \varphi _1\bigg (\bigg [\frac{{1} }{\sqrt{n_0}}\bigg ]_i\bigg ) + Z_i^{^{\prime }}\bigg )\\
\cdot \bigg (X_j^0\varphi _1^{^{\prime ... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.02078116312623024,
0.052486930042505264,
-0.002199034672230482,
0.004810030572116375,
-0.009169954806566238,
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-0.003917447291314602,
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0.0033948668278753757,
0.0035321873147040606,
-0.0461396723985672,
-0.025251707062125206,
-0.0007495408062823117... |
eee3f1be410b0817aceb32b8f63be3c47546839f | subsection | 125 | 160 | Concentration with respect to | First notice that\frac{\partial \Gamma _{t, \epsilon ,\mu }}{\partial ({1})_{ij}}\\= \sqrt{\frac{1-t}{n_0 n_1}}({2})_{\mu i}
\bigg (
X_j^0 \varphi _1^{^{\prime }}\bigg (\bigg [\frac{{1} ^0}{\sqrt{n_0}}\bigg ]_i\bigg )
\varphi _2^{^{\prime }}\bigg (\sqrt{\frac{1-t}{n_1}}\, \bigg [{2}
\varphi _1\bigg (\frac{{1} ^0}{\sqrt... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.038825783878564835,
0.03849002718925476,
-0.01863454468548298,
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-0.01568903587758541,
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0.034155700355768204,
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0.018436143174767494,
0.013140330091118813,
-0.025303911417722702,
-0.011392864398658276,
-0.030996527522802353,
... |
e7af0d4bf8622e4f541fd1541d724ebc00eccf18 | subsection | 126 | 160 | Concentration with respect to | \widehat{\mathcal {H}}_{t,\epsilon }} \bigg ],where\tilde{\Gamma }_{t, \epsilon ,\mu }^{(ij)} X_j^0 \varphi _1^{^{\prime }}\bigg (\bigg [\frac{{1} ^0}{\sqrt{n_0}}\bigg ]_i\bigg )
\varphi _2^{^{\prime }}\bigg (\sqrt{\frac{1-t}{n_1}}\, \bigg [{2}
\varphi _1\bigg (\frac{{1} }{\sqrt{n_0}}\bigg )
\bigg ]_\mu + k_1(t) \,V_{\... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.01117180846631527,
0.05332554504275322,
-0.021443158388137817,
-0.0009591234265826643,
-0.005379866808652878,
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0.011507573537528515,
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0.024205585941672325,
0.003672426799312234,
-0.02197732962667942,
-0.018879136070609093,
0.011881493031978607,
... |
0fc7d0b227e283dc04311be8ccb6d02ae95ef607 | subsection | 127 | 160 | Concentration with respect to | ({2})_{\mu i} gives&\tilde{}\bigg [ ({2})_{\mu i} \bigg \langle \!\!(\Gamma _{t,\epsilon ,\mu } + \sqrt{\Delta }Z_\mu ) \tilde{\Gamma }_{t,\epsilon ,\mu }^{(ij)} \bigg \rangle _{\!\!\widehat{\mathcal {H}}_{t,\epsilon }}\bigg ]\\
&\qquad \qquad \qquad = \tilde{}\bigg [ \bigg \langle \frac{\partial \Gamma _{t, \epsilon ,... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.02934139221906662,
0.026869574561715126,
-0.05230488255620003,
-0.03549041971564293,
-0.0036333431489765644,
-0.013915418647229671,
0.010451821610331535,
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0.021742841228842735,
0.007445969618856907,
-0.04903964325785637,
0.02369588240981102,
-0.017211174592375755,
... |
8fea264ce61909829ce3b73511b28938c75129e1 | subsection | 128 | 160 | Concentration with respect to | \widehat{\mathcal {H}}_{t,\epsilon }} \,\bigg ].The first two conditional expectations satisfy\Bigg \vert \tilde{}\bigg [ \bigg \langle \frac{\partial \Gamma _{t, \epsilon ,\mu }}{\partial ({2})_{\mu i}} \tilde{\Gamma }_{t, \epsilon , \mu }^{(ij)} \bigg \rangle _{\!\! \widehat{\mathcal {H}}_{t,\epsilon }} \,\bigg ] \Bi... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.017380330711603165,
0.04300067573785782,
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0.00850705336779356,
0.010902762413024902,
-0.0367443673312664,
0.011703874915838242,
-0.0009527518996037543,
... |
11a055963ec51ad55280075ac7e91b473a3dbb35 | subsection | 129 | 160 | Concentration with respect to | \widehat{\mathcal {H}}_{t,\epsilon }} \,\bigg ] \Bigg \vert &\le \bigg ( 2\sup \vert \varphi _2\vert + \sqrt{\frac{2\Delta }{\pi }} \bigg ) \frac{2S \sup \vert \varphi _1^{^{\prime }} \vert \sup \vert \varphi _1 \vert \sup \vert \varphi _2^{^{\prime \prime }} \vert }{\sqrt{n_1}} \,,while for the last two we have&\bigg ... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.03513291850686073,
0.04016934707760811,
-0.008493253029882908,
0.0017465348355472088,
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0.01799379289150238,
0.028356628492474556,
-0.04914335161447525,
0.02960810624063015,
0.005879650823771954,
0.... |
f768243bc36432ac9d27b3a671560fec8f6adb21 | subsection | 130 | 160 | Concentration with respect to | \widehat{\mathcal {H}}_{t,\epsilon }} \bigg \langle \big (\Gamma _{t, \epsilon ,\mu } + \sqrt{\Delta }Z_\mu \big ) \frac{\partial \Gamma _{t, \epsilon ,\mu }}{\partial ({2})_{i j}} \bigg \rangle _{\!\! \widehat{\mathcal {H}}_{t,\epsilon }}
\,\bigg ] \bigg \vert \\
&\qquad \le \Bigg ( 4 \sup \vert \varphi _2\vert ^2 + \... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.02365362085402012,
0.033145591616630554,
-0.03699121251702309,
-0.01996060460805893,
-0.000950913701672107,
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0.024874452501535416,
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0.01623706705868244,
-0.04105047881603241,
-0.003338212613016367,
0.020738884806632996,
0... |
0ba997002e1434c885308be7fe80a8705050b0cb | subsection | 131 | 160 | Concentration with respect to | Finally\big \Vert \nabla g \big \Vert ^2
= \sum _{i=1}^{n_1} \sum _{j=1}^{n_0} \bigg \vert \frac{\partial g}{\partial ({1})_{i j}} \bigg \vert ^2
\le \frac{1}{n_0} \cdot \frac{n_1}{n_0} \Big ( C_2(\varphi _1, \varphi _2, \alpha _1, \alpha _2, S \Big )^2a.s. and an application of Proposition REF ends the proof.
Under ... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.014217621646821499,
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-0.019292617216706276,
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0.009989729151129723,
0.003897826187312603,
0... |
221445f4ad99e4f4510d7d12d4cbfbab0cec7f1f | subsection | 132 | 160 | Concentration with respect to | Also we omit the additional terms k_1(t) \,+ k_2(t) \,/ that appear in the first argument of \varphi _2, To be clear, we will abusively write:\varphi _1\bigg ( \frac{{1} ^0}{\sqrt{n_0}}\bigg ) \equiv \varphi _1\bigg ( \frac{{1} ^0}{\sqrt{n_0}}, _1\bigg )\,,\;
\varphi _1\bigg ( \frac{{1} }{\sqrt{n_0}}\bigg ) \equiv \var... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.02789466828107834,
0.057559460401535034,
-0.008469115011394024,
-0.01457298081368208,
-0.03041251376271248,
-0.03909526392817497,
0.020371653139591217,
0.00016678338579367846,
0.024613840505480766,
0.02020379714667797,
-0.02827616035938263,
-0.011254003271460533,
-0.018357377499341965,
... |
93d896d719d6aba50cf29feacd49871081e3c27f | subsection | 133 | 160 | Concentration with respect to | For j \in \lbrace 1,\dots ,n_0\rbrace :\frac{\partial g}{\partial X_j^0}
=\\ \frac{-\Delta ^{-1}}{n_0^{{3}{2}} \sqrt{n_1}} \sum _{\mu =1}^{n_2} \sum _{i=1}^{n_1}
\tilde{}\bigg [({1})_{ij}({2})_{\mu i}\varphi _1^{^{\prime }}\bigg (\bigg [\frac{{1} ^0}{\sqrt{n_0}}\bigg ]_i\bigg )
\varphi _2^{^{\prime }}\bigg (\sqrt{\frac... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.05014510825276375,
0.013795246370136738,
-0.011460430920124054,
-0.04092793166637421,
0.003393493127077818,
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0.017320359125733376,
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0.007653004489839077,
-0.028139861300587654,
-0.00739358039572835,
-0.01896846294403076,
-0... |
9b82ff037121a220ee9f83d24855830bdf3f6c05 | subsection | 134 | 160 | Concentration with respect to | \widehat{\mathcal {H}}_{t,\epsilon }} \, \bigg ]\,.This expression can be further simplified using that ,^{\prime } are centred and independent of everything:\frac{\partial g}{\partial X_j^0}=\\
\frac{-\Delta ^{-1}}{n_0 \sqrt{n_0 n_1}} \sum _{\mu =1}^{n_2} \sum _{i=1}^{n_1}
\tilde{}\bigg [({1})_{ij}({2})_{\mu i}\varphi... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.001606068224646151,
0.037813179194927216,
-0.022858815267682076,
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0.00988819170743227,
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0.018082574009895325,
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0.03961380571126938,
0.016205647960305214,
-0.05197404697537422,
-0.007595444098114967,
-0.01031546015292406,
-... |
62f0e4345ed9a2810c70689736f01fe42de73599 | subsection | 135 | 160 | Concentration with respect to | 2} \, \bigg ] \,,whose absolute value is upperbounded by {C(\varphi _1, \varphi _2, \alpha _1, \alpha _2, S)}{\sqrt{n_0}}, and&\tilde{}\bigg [({1})_{ij}
\varphi _1^{^{\prime }}\bigg (\bigg [\frac{{1} ^0}{\sqrt{n_0}}\bigg ]_i\bigg )
\bigg \langle \varphi _1\bigg (\bigg [\frac{{1} }{\sqrt{n_0}}\bigg ]_i\bigg )\bigg \rang... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.007856498472392559,
0.044911712408065796,
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0.015461284667253494,
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0.0007007920648902655,
... |
3feec4c6310b7d8cc8cc21f5158d8017a95ed79b | subsection | 136 | 160 | Concentration with respect to | \widehat{\mathcal {H}}_{t,\epsilon }} \bigg ]\\
&\qquad \qquad \qquad \qquad \qquad \qquad \qquad + \tilde{}\bigg [
\varphi _1^{^{\prime }}\bigg (\bigg [\frac{{1} ^0}{\sqrt{n_0}}\bigg ]_i\bigg )
\bigg \langle \varphi _1\bigg (\bigg [\frac{{1} }{\sqrt{n_0}}\bigg ]_i\bigg ) \bigg \rangle _{\!\! \widehat{\mathcal {H}}_{t,... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.009391951374709606,
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0.009613207541406155,
... |
c56a9ac2ce7f74e8771d417edbbfae6f4a915499 | subsection | 137 | 160 | Concentration with respect to | \widehat{\mathcal {H}}_{t,\epsilon }} \bigg ] \bigg \vert \le \frac{C(\varphi _1, \varphi _2, \alpha _1, \alpha _2, S)}{\sqrt{n_0}} \,.It remains to upperbound, for every pair (\mu ,i) \in \lbrace 1,\dots ,n_2\rbrace \times \lbrace 1,\dots ,n_1\rbrace , the absolute value of the conditional expectation&\tilde{}\bigg [(... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.01448829099535942,
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0.026794563978910446,
0... |
e00f9183bcfd4568d74018f8ec2df459642876d8 | subsection | 138 | 160 | Concentration with respect to | ({1})_{ij} returns:&\tilde{}\bigg [({1})_{ij}({2})_{\mu i}\varphi _1^{^{\prime }}\bigg (\bigg [\frac{{1} ^0}{\sqrt{n_0}}\bigg ]_i\bigg )
\varphi _2^{^{\prime }}\bigg (\sqrt{\frac{1-t}{n_1}}\big [{2} ^1\big ]_{\mu }\bigg )
\bigg \langle \varphi _2\bigg (\sqrt{\frac{1-t}{n_1}}\big [{2} ^1\big ]_{\mu }\bigg ) \bigg \rangl... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.02400030940771103,
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-0.0025823600590229034,
0... |
550ac4946d0afbc44d1517c572532ca23cec25d9 | subsection | 139 | 160 | Concentration with respect to | \widehat{\mathcal {H}}_{t,\epsilon }}\,\bigg ]\\
&\;+\tilde{}\bigg [\sqrt{\frac{1-t}{n_0 n_1}}({2})_{\mu i}^2
\varphi _1^{\prime }\bigg (\bigg [\frac{{1} ^0}{\sqrt{n_0}}\bigg ]_i\bigg )
\varphi _2^{^{\prime }}\bigg (\sqrt{\frac{1-t}{n_1}}\big [{2} ^1\big ]_{\mu }\bigg )\\
&\;\qquad \qquad \qquad \qquad \qquad \qquad \q... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.006235354579985142,
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-0.0026981732808053493,
... |
9d2f23f00b7259a1f32239375d8f905fcb9570d4 | subsection | 140 | 160 | Concentration with respect to | \widehat{\mathcal {H}}_{t,\epsilon }}
\bigg \langle \frac{\partial \widehat{\mathcal {H}}_{t,\epsilon }}{\partial ({1})_{ij}} \bigg \rangle _{\!\! \widehat{\mathcal {H}}_{t,\epsilon }} \,\bigg ].The first conditional expectation on the right hand side can then be upperbounded, after integrating by parts w.r.t. ({2})_{\... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
0.014212372712790966,
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0.019163435325026512,
0.0... |
22a8c1c7c872e7c708348f7205717fa8fc44300a | subsection | 141 | 160 | Concentration with respect to | The second and third conditional expectations are easily upperbounded by\frac{1}{\sqrt{n_0 n_1}} \sup \vert \varphi _1^{^{\prime }}\vert ^2 \cdot \sup \vert \varphi _2\vert \cdot \sup \vert \varphi _2^{^{\prime \prime }}\vert \cdot \underbrace{\big [({2})_{\mu i}^2\big ]}_{= 1} \le \frac{C(\varphi _1, \varphi _2, \alph... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
0.002119450131431222,
0.03211033716797829,
0.0014202796155586839,
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0.015765199437737465,
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-0.003428129479289055,
0.01... |
e14baf549e3a99e45730ba4c2b45faea42a85cfa | subsection | 142 | 160 | Concentration with respect to | ({2})_{\mu i} and ({2})_{\nu i}\,;
the term due to the second line of (REF ) is upperbounded by {C(\varphi _1, \varphi _2, \alpha _1, \alpha _2, S)}{n_0} after integrating by parts w.r.t. ({2})_{\mu i}.All in all, there exists a positive constant C(\varphi _1, \varphi _2, \alpha _1, \alpha _2, S) such that almost sure... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.01436776202172041,
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0.024722011759877205,
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0.016130348667502403,
0.00... |
995fa8ecec54979f3fef8241776d493c7a5242b5 | subsection | 143 | 160 | Proof of Theorem | From Lemmas REF and REF , we directly obtain the bound{\mathbb {V}\mathrm {ar}}\bigg (\frac{\ln \widehat{}_{\mathbf {n},t,\epsilon }}{n_0} \bigg )
= \Bigg [\Bigg (\frac{\ln \widehat{}_{\mathbf {n},t,\epsilon }}{n_0} - \bigg [\frac{\ln \widehat{}_{\mathbf {n},t,\epsilon }}{n_0} \bigg \vert ^1, {1} \bigg ]\Bigg )^{\!\! 2... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.0062242355197668076,
0.04533928260207176,
-0.04155592620372772,
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0.017116647213697433,
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0.024424022063612938,
-0.03200599551200867,... |
1c03d906f0f14007badb79744d01d673e2d426a4 | subsection | 144 | 160 | Concentration of the overlap | This section presents the proof of Proposition REF . This proof is essentially the same as the one provided for the one-layer GLM . All along this section t \in [0,1] is fixed, and the averaged free entropy is treated as a mapping (R_1,R_2) \mapsto f_{\mathbf {n},\epsilon }(t) of R_1 = R_1(t,\epsilon ) and R_2 = R_2(t,... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.04162481427192688,
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0.0217279102653265,
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0.006500062998384237,
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0.008987176232039928,
-0.009277084842324257,
0.0... |
a7f484e09e2c69be2be8c1f0abcfe30b191d59e7 | subsection | 145 | 160 | Concentration of the overlap | Let\mathcal {L} \frac{1}{n_1} \sum _{i=1}^{n_1}\Bigg (
\frac{\big (x_i^1\big )^2}{2} - x_i^1 X_i^1 - \frac{x_i^1 Z_i^{\prime }}{2\sqrt{R_1}} \Bigg )\,.First we prove a formula that we uses extensively, in particular in Lemma REF .
[Formula for \langle \mathcal {L} \rangle _{\mathbf {n},t,\epsilon }]
For any \epsilon \i... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.026161646470427513,
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-0.02170730195939541,
... |
849411ab9e8d4c6076bebac0876039461eec3601 | subsection | 146 | 160 | Concentration of the overlap | From \mathcal {L} definition we directly get\langle \mathcal {L}\rangle _{\mathbf {n},t,\epsilon }
&= \frac{1}{n_1} \sum _{i=1}^{n_1} \frac{1}{2}\big [\big \langle (x_i^1)^2 \big \rangle _{t, \epsilon }\big ]
- \big [X_i^1 \big \langle x_i^1 \big \rangle _{t, \epsilon }\big ]
-\frac{1}{2\sqrt{R_1(t,\epsilon )}}\big [\b... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.045705635100603104,
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0.009862714447081089,
-0.01180779747664928... |
bad3d33c550fa7d6b272ddc0637377b1b36bf738 | subsection | 147 | 160 | Concentration of the overlap | \mathbf {n},t, \epsilon }\Big ]
\ge \frac{1}{4}\mathbb {E}\Big [\Big \langle \big (\widehat{Q} - \mathbb {E}\langle \widehat{Q} \rangle _{n, t, \epsilon }\big )^2\Big \rangle _{\! \mathbf {n},t,\epsilon }\Big ]\,.The full derivation in Section 6 of can be reproduced exactly – doing the identifications X_i^1 \leftrighta... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.024983681738376617,
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0.01704750955104828,
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0.02145818993449211,
-0.014170646667480469,
0.... |
e1d3515736c8656d641e88387a72b1b13b35754d | subsection | 148 | 160 | Concentration of the overlap | Under assumptions REF , REF , REF there exists a constant C(\varphi _1,\varphi _2,\alpha _1,\alpha _2, S) independent of t such that\int _{{\cal B}_{n_0}} \!\! d\epsilon \,\big [\big \langle \big (\mathcal {L} - \big [\langle \mathcal {L} \rangle _{\mathbf {n},t,\epsilon }\big ]\big )^2\big \rangle _{\mathbf {n},t,\eps... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.004680538084357977,
0.02859438769519329,
-0.050688665360212326,
0.017516732215881348,
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0.01067330501973629,
0.02029377594590187,
-0.007179114036262035,
-0.011062396690249443,
0.004287632182240486,
0.006992197595536709,
0.... |
9641cf2ec5b50bb664525d5241f524c2166724c6 | subsection | 149 | 160 | Concentration of the overlap | Under assumptions REF , REF , REF , we have for n_0 large enough\int _{{\cal B}_{n_0}} \!\! d\epsilon \,\big [\big \langle \big (\mathcal {L} - \langle \mathcal {L} \rangle _{\mathbf {n},t,\epsilon }\big )^2\big \rangle _{\mathbf {n},t,\epsilon }\big ]
\le \frac{\rho _1(1 + \rho _1)}{\alpha _{1} n_0} \,.For any realiza... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.0033764722757041454,
0.021395768970251083,
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0.013017539866268635,
0.001167458132840693,
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-0.015314304269850254,
-0.03534422442317009,... |
3b5030a64222e2348abd22f41d7030b255a04afb | subsection | 150 | 160 | Concentration of the overlap | 2} \big [\big (\langle \mathcal {L}^2 \rangle _{\mathbf {n},t,\epsilon } - \langle \mathcal {L} \rangle _{\mathbf {n},t,\epsilon }^2\big )\big ]
- \frac{1}{4n_0^2 R_1}\sum _{i=1}^{n_1}\big [\big \langle (x_i^1)^2 \big \rangle _{\mathbf {n},t,\epsilon } - \big \langle x_i^1 \big \rangle _{\mathbf {n},t,\epsilon }^2 \big... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.019480379298329353,
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... |
4144234ad80163be0402e9a77354944355595a4a | subsection | 151 | 160 | Concentration of the overlap | Integrating over \epsilon \in {\cal B}_{n_0} we obtain\int _{{\cal B}_{n_0}} \!\! d\epsilon \,\big [\big \langle \big (\mathcal {L} - \langle \mathcal {L} \rangle _{\mathbf {n},t,\epsilon }\big )^2\big \rangle _{\mathbf {n},t,\epsilon }\big ]\\
\le \frac{n_0}{n_1^2}\int _{R({\cal B}_{n_0})} \frac{dR_1dR_2}{J_R(R^{-1}(R... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
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c8c0f9778dcd62b4de72996d2ec53980b01bbaf3 | subsection | 152 | 160 | Concentration of the overlap | For the third inequality we made use of 0 \le -{df_{\mathbf {n},\epsilon }(t)}{dR_1} \le \frac{n_1}{2n_0}\rho _1(n_0). Finally, because s_{n_0} + s_{n_0}{\ln 2}{2} is less than 1 and \rho _1(n_0) \rightarrow \rho _1, we have\int _{{\cal B}_{n_0}} \!\! d\epsilon \,\big [\big \langle \big (\mathcal {L} - \langle \mathcal... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
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100eb1919ae749cedc4156db846f735a2dc6cb03 | subsection | 153 | 160 | Concentration of the overlap | Under assumptions REF , REF , REF , there exists a constant C(\varphi _1,\varphi _2,\alpha _1,\alpha _2, S) independent of t such that\int _{{\cal B}_{n_0}} \!\!\!\!\! d\epsilon \, \big [\big (\langle \mathcal {L}\rangle _{\mathbf {n},t,\epsilon } - \mathbb {E}\langle \mathcal {L}\rangle _{\mathbf {n},t,\epsilon }\big ... | {
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} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
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06117702619458ba73def6b5beb8840cbc50587a | subsection | 154 | 160 | Concentration of the overlap | From (REF ) we get\tilde{F}(R_1) - \tilde{f}(R_1)
= F_{\mathbf {n}, \epsilon }(t)
- f_{\mathbf {n}, \epsilon }(t) - \sqrt{R_1} \sup \vert \varphi _1 \vert \, \frac{n_1}{n_0} A \,,whose derivative reads – remember (REF ), (REF ) –\tilde{F}^{\prime }(R_1) - \tilde{f}^{\prime }(R_1)
= \frac{n_1}{n_0}\big (\big [\langle \m... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
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3ed722110a54dcf780b7be433996680724be1d57 | subsection | 155 | 160 | Concentration of the overlap | \delta ^{-1} \big (\big \vert \big (F_{\mathbf {n},\epsilon }(t) - f_{\mathbf {n},\epsilon }(t)\big )\big \vert _{R_1 = u}\big \vert + \sup \vert \varphi _1 \vert \, \frac{n_1}{n_0} \vert A \vert \sqrt{u} \big )\\
+ C_\delta ^+(R_1)
+ C_\delta ^-(R_1)
+\bigg \vert \frac{\Vert ^1\Vert ^2}{n_1}-\rho _1(n_0)\bigg \vert + ... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
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41efc45f066c1d3d933c54b4472af6eeab46467a | subsection | 156 | 160 | Concentration of the overlap | 2}\bigg ]
= \frac{1}{n_1^2}{\mathbb {V}\mathrm {ar}}\Bigg [\sum _{i=1}^{n_1}Z_i^{\prime } X_i^1\Bigg ]
= \frac{{\mathbb {V}\mathrm {ar}}[Z_1^{\prime } X_1^1]}{n_1}
\le \frac{[(X_1^1)^2]}{n_1} = \frac{\rho _1(n_0)}{n_1} \,,where we used that the random variables \lbrace Z_i^{\prime } X_i^1\rbrace _{1\le i \le n_1} are u... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
] | [
"cs.LG",
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"cs.IT",
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09d6f68c832039e4e0b722f47d3f5e75d2d4bdaf | subsection | 157 | 160 | Concentration of the overlap | This, combined with R_1 \ge \epsilon _1 \ge s_{n_0} gives\vert C_\delta ^{\pm }(R_1)\vert \le -\tilde{f}^{\prime }(R_1 - \delta )
\le \frac{n_1}{2 n_0} \sup \vert \varphi _1 \vert \bigg (1 + \frac{1}{\sqrt{R_1 - \delta }}\bigg )\!\! | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
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862200b9ed0e20ad6e766deb6783782187130551 | subsection | 158 | 160 | Concentration of the overlap | \underbrace{\le }_{n_0 \text{ large enough}} \!\!\!\!\!\alpha _1 \sup \vert \varphi _1 \vert \bigg (1 + \frac{1}{\sqrt{s_{n_0} - \delta }}\bigg ).Therefore, for n_0 large enough:&\int _{{\cal B}_{n_0}} \!\!\!\!\!\! d\epsilon \, \big (C^+(R_1(t,\epsilon ))^2 + C^-(R_1(t,\epsilon ))^2\big )&\quad \le \alpha _1 \sup \vert... | {
"cite_spans": []
} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
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d7cc49f9f2595b2656f1aefa219b7632f681be0d | subsection | 159 | 160 | Concentration of the overlap | 2}(s_{n_0} + \sup \vert \varphi _1 \vert )\,.In the later, the supremum \sup _{R_1} \vert \tilde{f}^{\prime }(R_1) \vert is taken over [s_{n_0}-\delta ,2s_{n_0}+\rho _1(n_0)+\delta ] and its upper bound \alpha _1 \sup \vert \varphi _1 \vert \Big (1 + \frac{1}{\sqrt{s_{n_0} - \delta }}\Big ) is uniform in R_2.
Integrati... | {
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} | 10.1088/1742-5468/ab3430 | 1805.09785 | Entropy and mutual information in models of deep neural networks | [
"Marylou Gabrié",
"Andre Manoel",
"Clément Luneau",
"Jean Barbier",
"Nicolas Macris",
"Florent Krzakala",
"Lenka Zdeborová"
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4d7c7ffd8eb0dda79bb59a8680f28e2c5753c1fe | abstract | 0 | 140 | Abstract | We study the classical Node-Disjoint Paths (NDP) problem: given an undirected
$n$-vertex graph G, together with a set {(s_1,t_1),...,(s_k,t_k)} of pairs of
its vertices, called source-destination, or demand pairs, find a
maximum-cardinality set of mutually node-disjoint paths that connect the demand
pairs. The best cur... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
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57cc6706b14f3932bc5e7f80e87cadabfa781451 | abstract | 1 | 140 | Abstract | We generalize this result to instances where the source vertices lie
within a prescribed distance from the grid boundary.
Much of the work on approximation algorithms for NDP relies on the
multicommodity flow relaxation of the problem, which is known to have an
$\Omega(\sqrt n)$ integrality gap, even in grid graphs. O... | {
"cite_spans": []
} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
] | [
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711099cb3b2735ac71f72881fb39fe194a031451 | subsection | 2 | 140 | Introduction | We study the classical Node-Disjoint Paths (NDP) problem, where the input consists of an undirected n-vertex graph G and a collection {\mathcal {M}}=\left\lbrace (s_1,t_1),\ldots ,(s_k,t_k) \right\rbrace of pairs of its vertices, called source-destination or demand pairs. We say that a path P routes a demand pair (s_i,... | {
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"raw": "N. Robertson and P. D. Seymour. Outline of a disjoint paths algorithm. In Paths, Flows and VLSI-Layout. Springer-Verlag, 1990.",
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"Julia Chuzhoy",
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0b97dfcbd9620f5b43a0cee5a8777ada0baef126 | subsection | 3 | 140 | Introduction | On the other hand, if, for every demand pair, either the source or the destination lies at a distance at least \Omega (n^{1/4}) from the grid boundary, then the integrality gap of the multicommodity flow relaxation improves, and one can obtain an \tilde{O}(n^{1/4})-approximation via LP-rounding. A natural question is w... | {
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the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
"Rachit Nimavat"
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6f20c0ac9c83825713cf8412f6afe80f634b0ba4 | subsection | 4 | 140 | Introduction | The best current approximation algorithm for Restricted NDP-Grid is the same as that for the general NDP-Grid, and achieves a \tilde{O}(n^{1/4})-approximation . Our main result is summarized in the following theorem.Theorem 1.1
There is an efficient randomized 2^{O(\sqrt{\log n}\cdot \log \log n)}-approximation algori... | {
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the Boundary | [
"Julia Chuzhoy",
"David H. K. Kim",
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d3993af08504e2a21f2d958ba5c5f4e02f751ecd | subsection | 5 | 140 | Introduction | As in the NDP problem, we can use the standard multicommodity flow LP-relaxation of the problem, in order to obtain the O(\sqrt{n})-approximation algorithm, and the integrality gap of the LP-relaxation is \Omega (\sqrt{n}) even in planar graphs.
Recently, Fleszar et al. designed an O(\sqrt{r}\cdot \log (kr))-approxima... | {
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"raw": "Krzysztof Fleszar, Matthias Mnich, and Joachim Spoerhase. New Algorithms for Maximum Disjoint Paths Based on Tree-Likeness. In Piotr Sankowsk... | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
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4d51987e6c01a45b61379666b1782276ee768840 | subsection | 6 | 140 | Other related work. | Cutler and Shiloach studied an even more restricted version of NDP-Grid, where all source vertices lie on the top row R^* of the grid, and all destination vertices lie on a single row R^{\prime } of the grid, far enough from its top and bottom boundaries. They considered three different settings of this special case. ... | {
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"Julia Chuzhoy",
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6be4c656c541b5d21b768eb55be38f1940e2fdbb | subsection | 7 | 140 | Organization. | We start with a high-level intuitive overview of our algorithm in Section . We then provide Preliminaries in Section and the algorithm for Restricted NDP-Grid in Section , with parts of the proof being deferred to Sections –. We extend our algorithm to EDP and NDP on wall graphs in Section . We generalize our algorith... | {
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} | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
the Boundary | [
"Julia Chuzhoy",
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f2f474c79528b7bd812a30458c53551422d7c7bc | subsection | 8 | 140 | High-Level Overview of the Algorithm | The goal of this section is to provide an informal high-level overview of the main result of the paper – the proof of Theorem REF .
With this goal in mind, the values of various parameters are given imprecisely in this section, in a way that best conveys the intuition. The following sections contain a formal descriptio... | {
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"raw": "Julia Chuzhoy and David H. K. Kim. On approximating node-disjoint paths in grids. In Naveen Garg, Klaus Jansen, Anup Rao, and Jo... | 1805.09956 | Improved Approximation for Node-Disjoint Paths in Grids with Sources on
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e45b7393711d947567e567457d4c180c0e221dae | subsection | 9 | 140 | High-Level Overview of the Algorithm | We call this part of the routing global routing. The local routing needs to specify how the paths in {\mathcal {P}} traverse each box B_i. This is done in a straightforward manner, while ensuring that the unique path originating at vertex s_i visits the vertex t_i (see Figure REF ). By suitably truncating the final set... | {
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