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bddf07f7d59fcb0aa9c6e3173611a43f94b03d1f | subsection | 7 | 37 | Wrapping up | In Narcissus, users specify formats using a library of
combinators, and use tactics to automatically derive
correct-by-construction encoder and decoder functions from these
specifications. Formats may be underspecified, in that a particular
source value may be serialized in different ways, but decoders are
guaranteed t... | {
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{
"arxiv_id": "",
"doi": "10.1007/978-3-642-00590-9_12",
"end": 2201,
"openalex_id": "https://openalex.org/W1509784638",
"raw": "Aditi Barthwal and Michael Norrish. 2009. Verified, Executable Parsing. In Programming Languages and Systems, Giuseppe Castagna (Ed.). Sp... | 1803.04870 | Narcissus: Deriving Correct-By-Construction Decoders and Encoders from
Binary Formats | [
"Benjamin Delaware",
"Sorawit Suriyakarn",
"Clément Pit--Claudel",
"Qianchuan Ye",
"Adam Chlipala"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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90245bd9840be423c2a28b9f0c81e639f13b9bbe | subsection | 8 | 37 | Narcissus, Formally | We begin our ground-up explanation of Narcissus with the
definition of the formats that capture relationships between
structured source values and their serialized representations. The
signature of a format from source type S to target type
|T| is defined by a type alias:
FormatM S T BIGSIGMA := Set of (S * BIGSIGMA *... | {
"cite_spans": []
} | 1803.04870 | Narcissus: Deriving Correct-By-Construction Decoders and Encoders from
Binary Formats | [
"Benjamin Delaware",
"Sorawit Suriyakarn",
"Clément Pit--Claudel",
"Qianchuan Ye",
"Adam Chlipala"
] | [
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f31a5b0c4618a3f5a0332b224f1ed9f54d2ca9c2 | subsection | 9 | 37 | Narcissus, Formally | The three operators have
straightforward interpretations as sets :
e ELEMENT return v === e = v
e ELEMENT x | P x === P e
e ELEMENT x <- y; k x === exists e'. e' ELEMENT y / e ELEMENT k e'As an example, we can specify the set of all possible
locations of a period in a string s as:
s1 <- s1 : String | exists s2. s = s... | {
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"raw": "Benjamin Delaware, Clément Pit-Claudel, Jason Gross, and Adam Chlipala. 2015. Fiat: Deductive Synthesis of Abstract Data Types in a Proof Assistant. In Proc. POPL.",
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Binary Formats | [
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"Clément Pit--Claudel",
"Qianchuan Ye",
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5ad328f74e41fbb6fdc41d380c92167270fa231c | subsection | 10 | 37 | Specifying Encoders and Decoders | These relational formats are not particularly useful by themselves—
even checking whether a format permits specific source and target
values may be undecidable. Instead we use them to specify the
correctness of both encoders and decoders.
So far, we have seen examples of relational formats that permit one or
many targe... | {
"cite_spans": []
} | 1803.04870 | Narcissus: Deriving Correct-By-Construction Decoders and Encoders from
Binary Formats | [
"Benjamin Delaware",
"Sorawit Suriyakarn",
"Clément Pit--Claudel",
"Qianchuan Ye",
"Adam Chlipala"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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e51d50b419682b9eccb274b95b722f1fd8aeb901 | subsection | 11 | 37 | Specifying Encoders and Decoders | For now, we require correct decoders
to flag strictly all malformed target
values by signaling errors when applied to target values not
included in the relation:
\forall s \, t, \; \mathsf {decode}\; t = \mathsf {Some}\; s
\rightarrow (s, t) \in \mathsf {format}. As we shall see later, our
formulation will also support... | {
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"doi": "",
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"raw": "P. Mockapetris. 1987. Domain names - implementation and specification. RFC 1035.",
"source_ref_id": "fa897e93f3ceaed0b4d52159bfbafc8bc39d9758",
"start": ... | 1803.04870 | Narcissus: Deriving Correct-By-Construction Decoders and Encoders from
Binary Formats | [
"Benjamin Delaware",
"Sorawit Suriyakarn",
"Clément Pit--Claudel",
"Qianchuan Ye",
"Adam Chlipala"
] | [
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8db350c4e3b37c8ba0cc323728b87a1df11ee3d2 | subsection | 12 | 37 | Specifying Encoders and Decoders | Hence our full notion
of decoder correctness accounts for both state and erroneous
target values.[Decoder Correctness]
A correct decoder for a format,
|format : FormatM S T BIGSIGMAE|, and relation on states,
|EQUIV : Set of (BIGSIGMAE * BIGSIGMAD)|, is a function,
|decode : DecodeM S T BIGSIGMAD|, that, when applied t... | {
"cite_spans": []
} | 1803.04870 | Narcissus: Deriving Correct-By-Construction Decoders and Encoders from
Binary Formats | [
"Benjamin Delaware",
"Sorawit Suriyakarn",
"Clément Pit--Claudel",
"Qianchuan Ye",
"Adam Chlipala"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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... | |
2d19609d7be99acb89dab6f906be96cc18ae509c | subsection | 13 | 37 | Specifying Encoders and Decoders | \;
\mathsf {decode}\; t\;\sigma _D = \mathsf {Some}(s, \sigma _D^{\prime })[Encode Inverts Decode]
Given a correct decoder
\mathsf {format} \hspace{-2.84544pt}\begin{}[baseline=-0.63ex, ->, thick]\node (A) { };\node (B) [right of=A, node distance=3em] { };\node (C) [right of=A, node distance=1.5em] {\approx };[every n... | {
"cite_spans": []
} | 1803.04870 | Narcissus: Deriving Correct-By-Construction Decoders and Encoders from
Binary Formats | [
"Benjamin Delaware",
"Sorawit Suriyakarn",
"Clément Pit--Claudel",
"Qianchuan Ye",
"Adam Chlipala"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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-0.019437817856669426... | |
77f16bb4ec124861e6e3dda81764eecd7c1746e7 | subsection | 14 | 37 | Deriving Encoders and Decoders | Equipped with precise notions of correctness, we can now define how we
derive provably correct encoders and decoders from a format. These
functions will be byte-aligned in a subsequent derivation step
presented in sec:ByteAlignment. We begin with encoders, since
they often have similar structure to their corresponding
... | {
"cite_spans": []
} | 1803.04870 | Narcissus: Deriving Correct-By-Construction Decoders and Encoders from
Binary Formats | [
"Benjamin Delaware",
"Sorawit Suriyakarn",
"Clément Pit--Claudel",
"Qianchuan Ye",
"Adam Chlipala"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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0.0... | |
61a59131de7e45504890c41a5e9b6ed0c41e981e | subsection | 15 | 37 | Decoders | Before defining similar correctness rules for decoder
combinators, we pause to consider how they are used to build a
top-level decoder. In particular, consider what the decoder
combinators used to build a reusable decoder for |++| should
look like:\mathsf {format_1 +\hspace{-3.0pt}+\, format_2}~ \hspace{-2.84544pt}\beg... | {
"cite_spans": []
} | 1803.04870 | Narcissus: Deriving Correct-By-Construction Decoders and Encoders from
Binary Formats | [
"Benjamin Delaware",
"Sorawit Suriyakarn",
"Clément Pit--Claudel",
"Qianchuan Ye",
"Adam Chlipala"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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0.0008098846301436424,
-0.004658983089029789,
-0.... | |
35b3abeb28638f61b9a0bef4d356de63b21c4eaa | subsection | 16 | 37 | Decoders | In order
to account for the first two concerns, our adaptation of the soundness
(left-inverse) criterion for decoder combinators is parameterized over
a binary relation on source and projected values representing a
view, as well as an additional format capturing the conformance
checking performed by the decoder.[Decode... | {
"cite_spans": []
} | 1803.04870 | Narcissus: Deriving Correct-By-Construction Decoders and Encoders from
Binary Formats | [
"Benjamin Delaware",
"Sorawit Suriyakarn",
"Clément Pit--Claudel",
"Qianchuan Ye",
"Adam Chlipala"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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3af2fb14a77bc6ed3232de4f924f4d5d25f023ab | subsection | 17 | 37 | Decoders | Absent a complete view of the original source
value, a combinator will be unable to ensure adherence to the original
format, but it can ensure that any computed value agrees with the
provided view format relation and is a consistent
view of any source values in the original format with the same encoding:
[Decoder Combi... | {
"cite_spans": []
} | 1803.04870 | Narcissus: Deriving Correct-By-Construction Decoders and Encoders from
Binary Formats | [
"Benjamin Delaware",
"Sorawit Suriyakarn",
"Clément Pit--Claudel",
"Qianchuan Ye",
"Adam Chlipala"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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0.00756384385749697... | |
dd05752fcacaa53a3a6fd68e25951ef718783d79 | subsection | 18 | 37 | Decoders | This predicate is key to our
approach to the modular verification of decoder combinators: each of
these proofs use this predicate to thread information about previously
decoded data through a proof of correctness for a composite
decoder. Such information is necessary for a decoder combinator whose
correctness depends o... | {
"cite_spans": []
} | 1803.04870 | Narcissus: Deriving Correct-By-Construction Decoders and Encoders from
Binary Formats | [
"Benjamin Delaware",
"Sorawit Suriyakarn",
"Clément Pit--Claudel",
"Qianchuan Ye",
"Adam Chlipala"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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... | |
550c6027385cdeb774a922795e6252991250f4d7 | subsection | 19 | 37 | Decoders | While not particularly helpful
during derivations, this rule is useful for proving other derivation
rules.
DecViewDone generalizes DecDone to arbitrary
views of a source value. The second premise corresponds to the
decision procedure from DecDone — an empty conformance
format is one consequence of the source value pro... | {
"cite_spans": []
} | 1803.04870 | Narcissus: Deriving Correct-By-Construction Decoders and Encoders from
Binary Formats | [
"Benjamin Delaware",
"Sorawit Suriyakarn",
"Clément Pit--Claudel",
"Qianchuan Ye",
"Adam Chlipala"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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fd849e4341435aaf91bd21d26c9a2f1bd573b271 | subsection | 20 | 37 | Improving Performance of Encoders and Decoders | The encoders and decoders derived via our combinator rules utilize the
same bitstring abstract data type as format specifications, employing
the bitstring's |snoc| and |unfold| operations
to enqueue and dequeue individual bits. Operating at the bit-level
imposes a large performance hit on these functions, since impleme... | {
"cite_spans": []
} | 1803.04870 | Narcissus: Deriving Correct-By-Construction Decoders and Encoders from
Binary Formats | [
"Benjamin Delaware",
"Sorawit Suriyakarn",
"Clément Pit--Claudel",
"Qianchuan Ye",
"Adam Chlipala"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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bff49b439a6815a054159a67f3528c158210eff8 | subsection | 21 | 37 | Improving Performance of Encoders and Decoders | We
define the twin equivalences used to justify the correctness of
byte-optimized functions as follows:
[Correctness of Byte-Aligned Encoders]
A byte-aligned encoder |encodebytes| and bit-aligned
encoder |encodebits| are equivalent,
\mathsf {encode\_bits \simeq encode\_bytes}, iff:|encodebytes| encodes the same bit seq... | {
"cite_spans": []
} | 1803.04870 | Narcissus: Deriving Correct-By-Construction Decoders and Encoders from
Binary Formats | [
"Benjamin Delaware",
"Sorawit Suriyakarn",
"Clément Pit--Claudel",
"Qianchuan Ye",
"Adam Chlipala"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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57a72e711d7fddacf5e5211a87b8e197d88996ad | subsection | 22 | 37 | Automating Derivations | As illustrated in sec:narcissusTour, Narcissus provides
a set of tactics to help automate the derivations described above. The
tactic derive_encoder_decoder_pair presented in that tour is
actually implemented via a pair of proof-automation tactics,
|DeriveEncoder| and |DeriveDecoder|, that derive
encoders and decoders,... | {
"cite_spans": []
} | 1803.04870 | Narcissus: Deriving Correct-By-Construction Decoders and Encoders from
Binary Formats | [
"Benjamin Delaware",
"Sorawit Suriyakarn",
"Clément Pit--Claudel",
"Qianchuan Ye",
"Adam Chlipala"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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0.... | |
6b9f53d0e5eebd87b6f85bff151b52e4abffc7e2 | subsection | 23 | 37 | Automating Derivations | The
subformats represent the \begin{}[baseline=-0.63ex, ->, thick]\node (A) { };\node (B) [right of=A, node distance=3em] { };\node (C) [right of=A, node distance=1.5em] {\approx };
\end{}[every node/.style={font=\sffamily \small }]
(A) edge[bend left=20] node [right] {} (B)
(B) edge[bend left=20] node [left] {} (A);
p... | {
"cite_spans": []
} | 1803.04870 | Narcissus: Deriving Correct-By-Construction Decoders and Encoders from
Binary Formats | [
"Benjamin Delaware",
"Sorawit Suriyakarn",
"Clément Pit--Claudel",
"Qianchuan Ye",
"Adam Chlipala"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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-0.009994063526391983... | |
29093e52ed89507ad2971017a75609d085223a2a | subsection | 24 | 37 | Automating Derivations | Automatically finding an instance of
\mathsf {s} is particularly worrisome, as it is well-known that
Ltac, Coq's proof-automation language, does not provide good support
for introspecting into definitions of inductive types, and we would
like to use Ltac to construct records of fairly arbitrary types,
without relying o... | {
"cite_spans": []
} | 1803.04870 | Narcissus: Deriving Correct-By-Construction Decoders and Encoders from
Binary Formats | [
"Benjamin Delaware",
"Sorawit Suriyakarn",
"Clément Pit--Claudel",
"Qianchuan Ye",
"Adam Chlipala"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
-0.03812701255083084,
0.006410466507077217,
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0.013454348780214787,
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0.0... | |
f0194821c12287399674077ae7b8511c29c8215a | subsection | 25 | 37 | Extending the Framework | As outlined in sec:narcissusTour, an extension to
Narcissus consists of four pieces: a format, encoder and decoder
combinators, derivation rules, and automation for incorporating these
rules into DeriveEncoder and DeriveDecoder. As a
concrete example, consider the format of the Internet Protocol (IP)
checksum used in t... | {
"cite_spans": []
} | 1803.04870 | Narcissus: Deriving Correct-By-Construction Decoders and Encoders from
Binary Formats | [
"Benjamin Delaware",
"Sorawit Suriyakarn",
"Clément Pit--Claudel",
"Qianchuan Ye",
"Adam Chlipala"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
-0.008482773788273335,
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0.000951165275182575,
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0.05364286154508591,
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0.003282116260379553,
-0.0017249704105779529,... | |
b3fbf4c29237f28297ab5249f9e955869f61ed7f | subsection | 26 | 37 | Evaluation | To evaluate the expressiveness and real-world applicability of Narcissus,
we wrote specifications and derived implementations of encoders and decoders for
five of the most commonly used packet formats of the Internet protocol suite:
Ethernet, ARP, IPv4, TCP, UDP. These formats were chosen to cover the full
TCP/IP stack... | {
"cite_spans": []
} | 1803.04870 | Narcissus: Deriving Correct-By-Construction Decoders and Encoders from
Binary Formats | [
"Benjamin Delaware",
"Sorawit Suriyakarn",
"Clément Pit--Claudel",
"Qianchuan Ye",
"Adam Chlipala"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
-0.003382350318133831,
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0.009401140734553337,
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-0.... | |
d24cfb866bf7961920ae986e260f39f96eb61732 | subsection | 27 | 37 | Benchmarking | fig:benchmarks shows single-packet encoding and decoding times,
estimated by linearly regressing over the time needed to run batches of n packet
serializations or deserializations for increasingly large values of n (complete
experimental data, including 95% confidence intervals, are provided as supplementary
material; ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 376,
"openalex_id": "",
"raw": "Christopher S. Hardin and Roshan P. James. 2013. Core_bench: micro-benchmarking for OCaml. (2013). https://github.com/janestreet/core_bench",
"source_ref_id": "5ed4250ef8dcef0a1bca024588311c5c... | 1803.04870 | Narcissus: Deriving Correct-By-Construction Decoders and Encoders from
Binary Formats | [
"Benjamin Delaware",
"Sorawit Suriyakarn",
"Clément Pit--Claudel",
"Qianchuan Ye",
"Adam Chlipala"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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-0.02402444928884506... | |
3123ceeb79cb33fdafc76f9bdeb3607d7ca36599 | subsection | 28 | 37 | Mirage OS Integration | MirageOS is a “library operating
system that constructs unikernels for secure, high-performance
network applications”: a collection of OCaml libraries that can be
assembled into a standalone kernel running on top of the Xen
hypervisor. Security is a core feature of MirageOS, making it a
natural target to demonstrate i... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 236,
"openalex_id": "",
"raw": "Anil Madhavapeddy, Richard Mortier, Charalampos Rotsos, David Scott, Balraj Singh, Thomas Gazagnaire, Steven Smith, Steven Hand, and Jon Crowcroft. 2013. Unikernels: Library Operating Systems for th... | 1803.04870 | Narcissus: Deriving Correct-By-Construction Decoders and Encoders from
Binary Formats | [
"Benjamin Delaware",
"Sorawit Suriyakarn",
"Clément Pit--Claudel",
"Qianchuan Ye",
"Adam Chlipala"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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3bb9636fda037b3b05107882f6f41c36e1ade760 | subsection | 29 | 37 | Setup | After extracting the individual encoders and decoders to OCaml, we
reprogrammed the TCP, UDP, IPv4, ARPv4, and Ethernet modules
of the mirage-tcpip library to use our
code optionally, and we recompiled everything. This whole
process went smoothly: Mirage's test suite did not reveal issues
with our proofs, though we did... | {
"cite_spans": []
} | 1803.04870 | Narcissus: Deriving Correct-By-Construction Decoders and Encoders from
Binary Formats | [
"Benjamin Delaware",
"Sorawit Suriyakarn",
"Clément Pit--Claudel",
"Qianchuan Ye",
"Adam Chlipala"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
-0.027047403156757355,
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0.005846541374921799,
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... | |
ac0dff06e3464d8d33ec79c29badc6ff0e9118dc | subsection | 30 | 37 | Parsers for Context-Free Languages | There is a long tradition of generating parsers for context-free
languages from declarative Backus-Naur-form
specifications , automatically.
Such generators may themselves have errors in them,
so in order to reduce the trusted code base of formally verified
compilers, there have been a number of efforts in verifying st... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 141,
"openalex_id": "",
"raw": "Stephen C. Johnson. 1979. Yacc: Yet Another Compiler-Compiler. Technical Report.",
"source_ref_id": "5169855be148f4cd6c6c5daa720ba3003b0ee701",
"start": 0
},
{
"arxiv_id": ... | 1803.04870 | Narcissus: Deriving Correct-By-Construction Decoders and Encoders from
Binary Formats | [
"Benjamin Delaware",
"Sorawit Suriyakarn",
"Clément Pit--Claudel",
"Qianchuan Ye",
"Adam Chlipala"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
-0.019225358963012695,
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6fc721cc19d02f47278e2b252edd0df9c34d49b9 | subsection | 31 | 37 | Verification of Parsers for Network Protocol Formats | A wide range of tools have been used to verify generated parsers for
binary network protocol formats , , , , , including the SAW symbolic-analysis
engine, , the Frama-C analyzer ,
F* , Agda , and Coq. The correctness
properties of each project differ from Narcissus's:
focus on memory safety. While and
prove that a p... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/978-3-319-96142-2_25",
"end": 201,
"openalex_id": "https://openalex.org/W2883286797",
"raw": "Nathan Collins, Mark Tullsen, Aaron Tomb, and Lee Pike. 2017. Formal Verification of a Vehicle-to-Vehicle (V2V) Messaging System. In Embed... | 1803.04870 | Narcissus: Deriving Correct-By-Construction Decoders and Encoders from
Binary Formats | [
"Benjamin Delaware",
"Sorawit Suriyakarn",
"Clément Pit--Claudel",
"Qianchuan Ye",
"Adam Chlipala"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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0.06024235114455223,
... | |
fdc540cf16e8e406e9675a9ed9502197e75cb2db | subsection | 32 | 37 | Deductive Synthesis | The idea of deriving correct-by-construction implementations from
specifications using deductive rules has existed for at least half a
century , . Kestrel's
Specware system was an seminal realization of this
idea, and has been used to implement correct-by-construction SAT
solvers , garbage collectors , and
network pro... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/bf01933419",
"end": 146,
"openalex_id": "https://openalex.org/W2078250023",
"raw": "Edsger W. Dijkstra. 1967. A constructive approach to the problem of program correctness. (Aug. 1967). http://www.cs.utexas.edu/users/EWD/ewd02xx/EWD... | 1803.04870 | Narcissus: Deriving Correct-By-Construction Decoders and Encoders from
Binary Formats | [
"Benjamin Delaware",
"Sorawit Suriyakarn",
"Clément Pit--Claudel",
"Qianchuan Ye",
"Adam Chlipala"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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... | |
5c8be60cc7d7aa4e32f94ddd0ca15b8692e9085b | subsection | 33 | 37 | Parser-Combinator Libraries | There is a long history in the functional-programming community of
using combinators to eliminate the burden of
writing parsers by hand, but less attention has been paid to the
question of how to generate both encoders and
decoders. Kennedy presents a library of
combinators that package serializers and deserializers ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 233,
"openalex_id": "https://openalex.org/W2123835026",
"raw": "Daan Leijen and Erik Meijer. 2001. Parsec: Direct style monadic parser combinators for the real world. (2001).",
"source_ref_id": "5f52b998f39eac493ae3b43695239... | 1803.04870 | Narcissus: Deriving Correct-By-Construction Decoders and Encoders from
Binary Formats | [
"Benjamin Delaware",
"Sorawit Suriyakarn",
"Clément Pit--Claudel",
"Qianchuan Ye",
"Adam Chlipala"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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005cc7fc1a4123bebccff07601a48374a603ed68 | subsection | 34 | 37 | Bidirectional / Invertible Programming Languages | Mu et. al present a functional language in which only injective
functions can be defined, allowing users to invert every program
automatically . The authors give a relational semantics
to this language, although every program in the language is a
function. The authors show how to embed noninjective programs in their
la... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/978-3-540-27764-4_16",
"end": 144,
"openalex_id": "https://openalex.org/W2115810994",
"raw": "Shin-Cheng Mu, Zhenjiang Hu, and Masato Takeichi. 2004. An Injective Language for Reversible Computation. In Mathematics of Program Constr... | 1803.04870 | Narcissus: Deriving Correct-By-Construction Decoders and Encoders from
Binary Formats | [
"Benjamin Delaware",
"Sorawit Suriyakarn",
"Clément Pit--Claudel",
"Qianchuan Ye",
"Adam Chlipala"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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0.04479518532752991,
... | |
257c463808626e836411071e854b17dd99d15cd4 | subsection | 35 | 37 | Extensible Format-Description Languages | Interface generators like XDR , ASN.1 , Apache
Avro , and Protocol Buffers generate
encoders and decoders from user-defined data schemes. The underlying
data format for these frameworks can be context-sensitive, but this
format is defined by the system, however, preventing data exchange
between programs using differen... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.17487/rfc4506",
"end": 36,
"openalex_id": "https://openalex.org/W1700387131",
"raw": "Raj Srinivasan. 1995. XDR: External data representation standard. Technical Report.",
"source_ref_id": "18c909cc2a81663eb7d85c0ed4d0d30448b71b39"... | 1803.04870 | Narcissus: Deriving Correct-By-Construction Decoders and Encoders from
Binary Formats | [
"Benjamin Delaware",
"Sorawit Suriyakarn",
"Clément Pit--Claudel",
"Qianchuan Ye",
"Adam Chlipala"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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1ccd3d047955bcb387d2514b85ff91b83c7a9312 | subsection | 36 | 37 | Conclusion | We have presented Narcissus, a framework for
specifying and deriving correct-by-construction encoders and decoders
for non-context-free formats in the style of parser-combinator
libraries. This framework provides fine-grained control over the shape
of encoded data, is extensible with user-defined formats and
implementa... | {
"cite_spans": []
} | 1803.04870 | Narcissus: Deriving Correct-By-Construction Decoders and Encoders from
Binary Formats | [
"Benjamin Delaware",
"Sorawit Suriyakarn",
"Clément Pit--Claudel",
"Qianchuan Ye",
"Adam Chlipala"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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828e02b8f4fa31cdad610ae1da33a6c132b44a61 | abstract | 0 | 160 | Abstract | We examine a class of deep learning models with a tractable method to compute
information-theoretic quantities. Our contributions are three-fold: (i) We show
how entropies and mutual informations can be derived from heuristic statistical
physics methods, under the assumption that weight matrices are independent and
ort... | {
"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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2889f71ed66cd43b6f03dfa2ba078fac54e7abf1 | subsection | 1 | 160 | Multi-layer model and main theoretical results | A stochastic multi-layer model—
We consider a model of multi-layer stochastic feed-forward neural network
where each element x_i of the input layer {x} \in \mathbb {R}^{n_0} is
distributed independently as P_0 (x_i), while hidden units t_{\ell , i} at each
successive layer {t}_\ell \in \mathbb {R}^{n_{\ell }} (vectors ... | {
"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... |
8c66becef99902cdf6646a8d890f9f6feb47fe35 | subsection | 2 | 160 | Multi-layer model and main theoretical results | Model
(REF ) thus describes a Markov chain which we denote by \rightarrow 1 \rightarrow 2 \rightarrow \dots \rightarrow L, with {\ell }=\varphi _{\ell }(W_{\ell }
{\ell -1}, { {\xi }}_\ell ), {\xi }_\ell =\lbrace {\xi }_{\ell ,i}\rbrace _{i=1}^{n_\ell }, and the activation function \varphi _\ell applied componentwise.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",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
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] | 2,018 | en | Computer Science | [
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fc998e9e159a595b5586cf68f06f993bb5bbc7c3 | subsection | 3 | 160 | Multi-layer model and main theoretical results | Then for any \ell \in \lbrace 1,\ldots ,L\rbrace the normalized entropy of \ell is given by the minimum among all
stationary points of the replica potential:\lim _{n_0 \rightarrow \infty } \frac{1}{n_0} H({\ell }) = \min _{{A}, {V}, {\tilde{A}}, {\tilde{V}}} \phi _\ell ({A}, {V}, {\tilde{A}}, {\tilde{V}}),which depends... | {
"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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18d1e0a01c18b359bd4756088f6b77de083a02cb | subsection | 4 | 160 | Multi-layer model and main theoretical results | In the computation of the conditional entropies in
(REF ), the scalar t_k-variables are generated from
P(t_0)=P_0(t_0) andP(t_{k} | \xi _k; A, V, \rho ) &= \mathbb {E}_{\tilde{\xi }, \tilde{z}}
\, P_k (t_k + \tilde{\xi } / \sqrt{A} |
\sqrt{\rho - V} \xi _k + \sqrt{V} \tilde{z}), \quad k=1,\dots ,\ell -1,\\
P(t_{\ell } ... | {
"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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-0.03837088495492935,
0.04... |
01d0bea128188b1152bf4daf2437a3b0f05da646 | subsection | 5 | 160 | Multi-layer model and main theoretical results | As ours, it exhibits layer-wise
additivity, and the two formulas are conjectured to be equivalent.Rigorous statement—
We recall the assumptions under which the replica formula of Claim
is conjectured to be exact: (i) weight matrices are
drawn from an ensemble of random orthogonally-invariant matrices,
(ii) matrices at... | {
"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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... |
cad5fc5ed633104ca12fc3e96185702bcd176289 | subsection | 6 | 160 | Tractable models for deep learning | The multi-layer model presented above can be leveraged to simulate two prototypical
settings of deep supervised learning on synthetic datasets amenable to the replica
tractable computation of entropies and mutual informations.
[Figure: Two models of synthetic data]The first scenario is the so-called teacher-student (se... | {
"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 | [
-0.03783775866031647,
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... |
0b2bfc773c1b5374373657f87881c3e640608239 | subsection | 7 | 160 | Tractable models for deep learning | In the following, we assume that is also the case for non-linear networks.In Section REF of the Supplementary Materialwe train a neural network with USV-layers on
a simple real-world dataset (MNIST), showing that these layers can learn to represent complex functions despite their restriction.
We further note that such ... | {
"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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... |
e2564c323564934532963af8d8ea1e45fb78f000 | subsection | 8 | 160 | Tractable models for deep learning | Additionally, they investigate
the training of larger linear networks on i.i.d. normally distributed inputs
where entropies at each hidden layer can be computed analytically for an
additive Gaussian noise. The strategy proposed in the present paper allows
us to evaluate entropies and mutual informations in non-linear n... | {
"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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-0.05946587398648262,
-0.022776436060667038,
... |
57f409eb5236536d09752579813092e500a4911a | subsection | 9 | 160 | Numerical experiments | We present a series of experiments both aiming at further
validating the replica estimator and leveraging its power in noteworthy
applications. A first application presented in the paragraph 3.1 consists
in using the replica formula in settings
where it is proven to be rigorously exact as a basis of comparison for othe... | {
"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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-0.04251305013895035... |
c857f2f9b94d00ca1a93ae5b52bb6199c1be2644 | subsection | 10 | 160 | Numerical experiments | To compute entropies, we consider noisy versions of
the latent variables where an additive white Gaussian noise of very
small variance (\sigma ^2_{\rm noise}=10^{-5}) is added right before
the activation function, 1 = f(W_1+ {\epsilon }_1) and
2 = f(W_2 f(W_1 ) + {\epsilon }_2) with
{\epsilon }_{1,2} \sim \mathcal {N}(... | {
"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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... |
7834a4f8db2537ed7bc422b384583ca2de2e4cb6 | subsection | 11 | 160 | Numerical experiments | This loss of information for bounded activations was also
observed by , where entropies were
computed by discretizing the output as a single neuron with bins of
equal size. In this setting, as the tanh activation starts to saturate
for large inputs, the extreme bins (at -1 and 1) concentrate more
and more probability 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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-0.009103639982640743,
-0.03546832874417305,... |
9fe04f3fda58ccdced5a9866a556cdfc626f24bf | subsection | 12 | 160 | Numerical experiments | The teacher has also to be linear to be learnable:
we consider a simple single-layer network with additive white Gaussian noise, =
\tilde{W}_{\rm teach} + {\epsilon }, with input {x} \sim \mathcal {N}(0, I_{n}) of size n, teacher matrix \tilde{W}_{\rm teach} i.i.d. normally distributed
as \mathcal {N}(0, 1/n) , noise {... | {
"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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-0.03808167949318886,
... |
a4fc27e7445111b58a93031a2e6bf3bbad92c3c8 | subsection | 13 | 160 | Numerical experiments | We then train a recognition model to
solve the binary classification problem of recovering the label y =
\mathrm {sign}(Y_1), the sign of the first neuron in , using plain
SGD but this time to minimize the cross-entropy loss. Note that the rest of
the initial code (Y_2, .. Y_{n_{}}) acts as noise/nuisance with respect ... | {
"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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01bfa333538ff73a55d015df015836af658c04c4 | subsection | 14 | 160 | Conclusion and perspectives | We have presented a class of deep learning models together with a
tractable method to compute entropy and mutual information between
layers. This, we believe, offers a promising framework for further
investigations, and to this aim we provide Python packages that
facilitate both the computation of mutual informations a... | {
"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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0.006542363669723272,
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... |
6e8c61894814ca66b68db5bb9b41b7631c472cd4 | subsection | 15 | 160 | Background | The replica method , was
first developed in the context of disordered physical systems where the
strength of interactions J are randomly distributed, J \sim P_J (J).
Given the distribution of microstates {x} at a fixed temperature
\beta ^{-1}, P ({x} | \beta , J) = \frac{1}{\mathcal {Z} (\beta , J)} \,
e^{-\beta \mathc... | {
"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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0.03051796369254589,
-0.03131143003702164,
0.02... |
67498b35c7c9c697b6373398a5020afc8fe65a3b | subsection | 16 | 160 | Background | Note this
quantity is nothing but the entropy of {y} given W, H({y} | W).The distribution P_J (or P_W in the notation above) is usually assumed
to be i.i.d. on the elements of the matrix J. However, one can also use
the same techniques to approach J belonging to arbitrary
orthogonally-invariant ensembles. This approach... | {
"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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0.0113160265609622,
0.005230705253779888,
0.017... |
24263ba79218a7ae675e9bc86608dc0ba4aff4ea | subsection | 17 | 160 | Single-layer | For a single-layer generalized linear model\left\lbrace \begin{aligned}&{x} \sim P_X ({x}), \\
&{y} \sim P_{Y | Z} ({y} | W {x}).
\end{aligned} \right.with P_X and P_{Y | Z} separable in the components
of {x} \in \mathbb {R}^n and {y} \in \mathbb {R}^m, and
W \in \mathbb {R}^{m \times n} Gaussian i.i.d., W_{\mu i} \sim... | {
"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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0.006213609129190445,... |
87cdbd2f0106f3d2e1edc42278016e7ddf1f2235 | subsection | 18 | 160 | Single-layer | The entropy is then written
as n^{-1} H ({y} | W) = \min _{A, V, \tilde{A}, \tilde{V}} \phi (A, V,
\tilde{A}, \tilde{V}), where\phi (A, V, \tilde{A}, \tilde{V}) = -\frac{1}{2} \big ( \tilde{A} V
+ \alpha A \tilde{V} - F_W(AV) \big ) + I(x; x +
\frac{\xi _0}{\sqrt{\tilde{A}}}) + \alpha H(y | \xi _1; \tilde{V},
\tilde{\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",
"cond-mat.dis-nn",
"cs.IT",
"math.IT",
"stat.ML"
] | 2,018 | en | Computer Science | [
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0.054447613656520844,
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-0.00441394979134202,
0.010063957422971725,
0.00... |
80097aec5e32864d0ece4ec4235c9b3fd213f067 | subsection | 19 | 160 | Multi-layer | Consider the following multi-layer generalized linear model\left\lbrace
\begin{aligned}&t_{0,i} \equiv x_i \sim P_0 (x_i), \\
&t_{1,i} \sim P_1 (t_{1,i} | W_1 {x}), \\
&t_{2,i} \sim P_2 (t_{2,i} | W_2 {t}_{1}), \\
&\vdots \\
&t_{L,i} \equiv y_i \sim P_L (y | W_L {t}_{L - 1}),
\end{aligned}
\right.where the W_\ell \in ... | {
"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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-0.02078564651310444,
0.0... |
764d3cadcff1463fe78d137ebc989532feea44b8 | subsection | 20 | 160 | Multi-layer | Our analysis employs the framework introduced in
to compute the entropy of {t}_L in the
limit n_0 \rightarrow \infty with \tilde{\alpha }_\ell = n_\ell / n_0 finite for
\ell = 1, \dots , L\lim _{n_0 \rightarrow \infty } n_0^{-1} H({t}_L | {W}) = \min _{{A}, {V}, {\tilde{A}}, {\tilde{V}}} \phi ({A},
{V}, {\tilde{A}}, {... | {
"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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0.013550326228141785,
0.0... |
ffb074c4c99ba015e3cfdd34d30ee5952b6eaa64 | subsection | 21 | 160 | A simple heuristic derivation of the multi-layer formula | Formula (REF ) can be derived using a simple argument. Consider
the case L = 2, where the model reads\left\lbrace \begin{aligned}&{t}_0 \sim P_0 ({t}_0), \\
&{t}_1 \sim P_1 ({t}_1 | W_1 {t}_0), \\
&{t}_2 \sim P_2 ({t}_2 | W_2 {t}_1),
\end{aligned} \right.with {t}_\ell \in \mathbb {R}^{n_\ell } and W \in \mathbb {R}^{n_... | {
"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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0.05408722534775734,
0.022053059190511703,
-0.03873402252793312,
-0.012461123056709766,
-0.005753635428845882,
... |
73aea1c1437ee84d7fe871b9d710d7de57b02ba3 | subsection | 22 | 160 | A simple heuristic derivation of the multi-layer formula | \end{aligned}Moreover, H(t_1 + \tilde{\xi }_1 / \sqrt{\tilde{A}}_2) can be obtained
from the replica free energy of another problem: that of estimating
{t}_0 given the knowledge of (noisy) {t}_1, which can again be
written using (REF )\lim _{n_0 \rightarrow \infty } n_0^{-1} \, H(t_1 +
\frac{\tilde{\xi }_1}{\sqrt{\tild... | {
"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 | [
-0.0428294911980629,
0.02297162637114525,
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0.009005487896502018,
0... |
b692d080b33f283bd4b33613da0da63b64c8d6e8 | subsection | 23 | 160 | Formulation in terms of tractable integrals | While expression (REF ) is more easily written in terms of
conditional entropies and mutual informations, evaluating it requires us
to explicitely state it in terms of integrals, which we do below.
We first consider the Gaussian i.i.d. In this case, the multi-layer
formula was derived with the cavity and replica method... | {
"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 | [
-0.04204684868454933,
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0.030909620225429535,
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0.024990107864141464,
-0.036371439695358276,
0.... |
9cb4ffcca445586a6a6807e9e7c45202722217a2 | subsection | 24 | 160 | Formulation in terms of tractable integrals | \end{aligned}where&Z_0 (A, B) \!=\! {\textstyle \int } dx \, P_0 (x) e^{-\frac{1}{2} A x^2 + B x}, \\
&Z_\ell (A, B, V, \omega ) \!=\! {\textstyle \int } dt dz \, P_\ell (t | z)
\mathcal {N} (z; \omega , V) e^{-\frac{1}{2} A t^2 + Bt}, \\
&Z_L (y, V, \omega ) \!=\! {\textstyle \int } dz \, P_L(y | z) \mathcal {N} (z; \... | {
"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 | [
-0.025691518560051918,
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0.03356374055147171,
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0.04341927915811539,
0.01582072675228119,
-0.000... |
2fc745149be8f45953d4163f0bc323ca814e1f35 | subsection | 25 | 160 | Recovering the formulation in terms of conditional entropies | One can rewrite the formulas above in a simpler way. By manipulating the
measures (REF ) one obtainsK_0 (A, \rho ) = -I(x; b) + \frac{1}{2} A \rho ,for x \sim P_0(x) and b \sim \mathcal {N} (b; Ax, A). Introducing a
standard normal variable \xi _0 and using the invariance of mutual
informations, this can be written asK... | {
"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 | [
-0.03965349867939949,
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0.02434462122619152,
0.003922612871974707,
... |
160769964679f2b37ae02c4de3e7a105e63b7db6 | subsection | 26 | 160 | Recovering the formulation in terms of conditional entropies | Introducing standard normal \xi _\ellK_\ell (A, V, \rho ) = -H(t_\ell | \xi _\ell ; A, V) + \frac{1}{2} A
\rho + \frac{1}{2} \log (2 \pi e A^{-1})\,,whereP(t_\ell | \xi _\ell ; A, V, \rho ) = \int \mathcal {D}\tilde{\xi } \mathcal {D}\tilde{z} \,
P_\ell (t_\ell + \sqrt{1 / A} \tilde{\xi } | \sqrt{\rho - V} \xi _\ell +
... | {
"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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0.03359433636069298,
-0.009603830054402351,
... |
c00f4229f795864537552f4749019bae15e96329 | subsection | 27 | 160 | Solving saddle-point equations | In order to deal with the extremization problem in\lim _{n_0 \rightarrow \infty } n_0^{-1} H ({t}_L | {W}) = \min _{{A},
{V}, \tilde{{A}}, \tilde{{V}}} \phi ({A}, {V},
\tilde{{A}}, \tilde{{V}}),one needs to solve the saddle-point equations \nabla _{\lbrace {A}, {V},
\tilde{{A}}, \tilde{{V}}\rbrace } \phi = 0. In what 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 | [
-0.03933761641383171,
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-0.022217055782675743,
0.006164622493088245,
... |
58f255ff22fcbe8bd68316aedfd550aec0d58f62 | subsection | 28 | 160 | Method 1: fixed-point iteration | We first introduce the following function, which is related to the
derivatives of F_{W_\ell }\psi _\ell (\theta , \gamma ) = 1 - \gamma \big [ \mathcal {S}_\ell \big ( -\gamma ^{-1}
(1 - \theta ) (1 - \alpha _\ell \theta ) \big ) \big ]^{-1},where \mathcal {S}_\ell (z) = \mathbb {E}_{\lambda _\ell }
\frac{1}{\lambda _\... | {
"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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0.02639472857117653,
0.0003213520103599876,
0.03710519149899483,
-0.0068504237569868565,
... |
265d46a87f1b866fc5db22a0f94dc9854ec31dc6 | subsection | 29 | 160 | Method 1: fixed-point iteration | After these quantities are computed for all
layers, we compute all the V_\ell ; for 2 \le \ell \le LV_\ell ^{(t + 1)} = \mathbb {E}_{b, t, z, w | \tilde{\rho }_{\ell - 1},
\tilde{A}_{\ell }, \tilde{V}_{\ell - 1}} \partial _b^2 \log Z_\ell (\tilde{A}_{\ell }^{(t)}, b, \tilde{V}_{\ell - 1}^{(t)}, w),and for the 1st layer... | {
"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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0.01276196725666523,
-0.009786971844732761,
0.... |
d59bcfc85ce72119bb2750a5e9c6c82534f0d6ab | subsection | 30 | 160 | Method 2: ML-VAMP state evolution | While the fixed-point iteration above works well in most cases, it is not
provably convergent. In particular, it relies on a solution for \theta =
\psi _\ell (\theta , A_\ell ^{(t)} V_\ell ^{(t)}) being found, which might not
happen throughout the iteration.An alternative is to employ the state evolution (SE) of the ML... | {
"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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-0.0016436834121122956,
0.010746867395937443,
... |
5530dee7d062887f7fd4ac88354622015e6b3650 | subsection | 31 | 160 | Method 2: ML-VAMP state evolution | Let us first look at
the single-layer case; the ML-VAMP SE equations read&A^+_x = \frac{1}{V^+_x (A^-_x)} - A^-_x, &&\qquad A^+_z = \frac{1}{V^+_z (A^+_x, 1 / A_z^-)} - A^-_z, \\
&A^-_x = \frac{1}{V^-_x (A^+_x, 1 / A_z^-)} - A^+_x, &&\qquad A^-_z = \frac{1}{V^-_z (A^+_z)} - A^+_z,where&V^+_x (A) = \mathbb {E}_{x, z} \p... | {
"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... |
be074e7e47cc147e1fc4691d8e7190240ba5ad51 | subsection | 32 | 160 | Mutual information from entropy | While in our computations we focus on the entropy H(\ell ), the mutual
information I(\ell ; {\ell - 1}) can be easily obtained from the chain
rule relationI(\ell ; {\ell -1}) &= H(\ell ) + {\ell }{\ell -1} \log P_{\ell | {\ell - 1}} ({t}_\ell | {t}_{\ell -1}) \\
&= H(\ell ) + \int dz \, \mathcal {N} (z; 0, \tilde{\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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0.008435932919383049,
... |
b29981b3cf8d7ed25260a9c87d25df4e0c9e6138 | subsection | 33 | 160 | Equivalence in linear case | In the linear case, = W_L W_{L - 1} \cdots W_1 + \mathcal {N} (0,
\Delta ), our formula reduces to
, ,\lim _{n_ \rightarrow \infty } \, n^{-1} I(; ) = \min _{A, V} \, \left\lbrace
-\frac{1}{2} A V - \frac{1}{2} G(-V / \Delta ) + I (x; x + \sqrt{1 / A} \xi ) \right\rbrace ,whereG(x) = _{\Lambda } \left\lbrace -\mathbb ... | {
"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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0.0076691280119121075,
0.0... |
e5c2645510904dc5a0f49696f3d69b4f7d577b31 | subsection | 34 | 160 | Proof of the replica formula by the adaptive interpolation method | In this section we prove Theorem (that we re-write below more explicitely) using the adaptive
interpolation method of , in a multi-layer
setting, as first developed in . | {
"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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-0.013967919163405895,... |
0bba10ba354c1f3bc9f40896386dbd47dee28f9b | subsection | 35 | 160 | Two-layer generalized linear estimation: Problem statement | One gives here a generic description of the observation model, that is a two-layer generalized linear model (GLM).
Let n_0, n_1, n_2 \in ^* and define the triplet \mathbf {n} = (n_0,n_1,n_2).
Let P_0 be a probability distribution over and let (X^0_i)_{i=1}^{n_0} P_0 be the components of a signal vector ^0.
One fixes tw... | {
"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... |
a19b77653aa0f99f66e43d481e196fb2e60136e1 | subsection | 36 | 160 | Two-layer generalized linear estimation: Problem statement | The estimation problem is to recover ^0 from the knowledge of =(Y_\mu )_{\mu =1}^{n_2}, \varphi _1, \varphi _2, {1}, {2} and \Delta , P_0.In the language of statistical mechanics, the random variables , {1}, {2}, ^0, _1, _2, are called quenched variables because once the measurements are acquired they have a “fixed rea... | {
"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 | [
-0.021127674728631973,
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0.0055145518854260445,
-0.013012207113206387,... |
77bc32012de8b140da8d5512931aede1dbdeb527 | subsection | 37 | 160 | Two-layer generalized linear estimation: Problem statement | The partition function is defined as(,{1}, {2}) \\
\!\int \!\! dP_0()dP_{A_1}(_1)dP_{A_2}(_2)
\prod _{\mu =1}^{n_2} \frac{1}{\sqrt{2\pi \Delta }}e^{-\frac{1}{2 \Delta }\Big ( Y_\mu - \varphi _2\Big (\frac{1}{\sqrt{n_1}} \Big [{2} \varphi _1\Big (\frac{{1} }{\sqrt{n_0}},_1\Big )\Big ]_{\mu },_{2,\mu }\Big )\Big )^2}.One... | {
"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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0.010077310726046562,
... |
0e1a8eafc0c20590af0949f3786f70f6d2cf7b14 | subsection | 38 | 160 | Important scalar inference channels | The thermodynamic limit of the free entropy will be expressed in terms of the free entropy of simple scalar inference channels. This “decoupling property” results from the mean-field approach in statistical physics, used through in the replica method to perform a formal calculation of the free entropy of the model , . ... | {
"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",
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] | 2,018 | en | Computer Science | [
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296061db4d2fd12b23a5e99780060f662afc850d | subsection | 39 | 160 | Important scalar inference channels | Suppose that V,U (0,1), where V is known. Consider the problem of recovering the unknown U from the observation Y_0^{\prime } = \sqrt{r}\varphi _1(\sqrt{q}\, V + \sqrt{\rho - q} \,U,_1) + Z^{\prime } where r \ge 0, \rho >0, q \in [0, \rho ], Z^{\prime } \sim (0,1) and _1 \sim P_{A_1}. Equivalently,
Y_0^{\prime } \sim P... | {
"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",
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] | 2,018 | en | Computer Science | [
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e497a27129dd2b0081be099adacdac03719640ee | subsection | 40 | 160 | Important scalar inference channels | Hence\Psi _{P_{\rm out,1}^{(r)}}(q;\rho ) = -\frac{1 + \ln (2\pi ) + r [\varphi _1^2(T,_1)]}{2} + \Psi _{\varphi _1}(q, r;\rho )where\Psi _{\varphi _1}(q, r;\rho ) \ln \int \! {\cal D}u dP_{A_1}() dh e^{\sqrt{r}h Y_0^{\prime } -\frac{r h^2}{2}} \delta \big (h -\varphi _1(\sqrt{q}\, V + \sqrt{\rho - q}\, u, )\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 | [
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0.... |
aed990e014587cc74d043981c68414bfee583f61 | subsection | 41 | 160 | Replica-symmetric formula and mutual information | Our goal is to prove Theorem that gives a single-letter replica-symmetric formula for the asymptotic free entropy
of model (REF ), (REF ). The result holds under the following hypotheses:The prior distribution P_0 has a bounded support.
\varphi _1, \varphi _2 are bounded \mathcal {C}^2 functions with bounded first a... | {
"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",
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] | 2,018 | en | Computer Science | [
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4e5158d889bb0a3394fe2190bfa57de0570ee289 | subsection | 42 | 160 | Interpolating estimation problem | The proof of Theorem REF follows the same steps than the proof of the replica formula for a one-layer GLM in . Let t\in [0,1] be an interpolation parameter. We introduce an interpolating estimation problem that interpolates between the original problem (REF ) at t=0 and two analytically tractable problems at t=1.Prior ... | {
"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.... |
b911d18b6054c3b8a6d0a22c9312cf420d6aa0a9 | subsection | 43 | 160 | Interpolating estimation problem | Let q_{\epsilon }: [0,1] \rightarrow [0,\rho _1(n_0)] and r_{\epsilon }: [0,1] \rightarrow [0,] be two continuous “interpolation functions”. Their dependence on \epsilon will also be specified later.
It is useful to defineR_1(t,\epsilon ) \epsilon _1 + \int _0^t r_{\epsilon }(v)dv\,,\qquad R_2(t,\epsilon ) \epsilon _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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... |
39cc1dbeed14f7754017ff1884a93b9af7da524c | subsection | 44 | 160 | Interpolating estimation problem | The inference problem is to estimate both unknowns ^0 and from the knowledge of , {1}, {2} and the two kinds of observations{\left\lbrace \begin{array}{ll}
Y_{t,\epsilon ,\mu } &\sim P_{\rm out, 2}(\ \cdot \ | \, S_{t,\epsilon ,\mu })\,, \qquad \qquad \qquad \qquad \;\;\, 1 \le \mu \le n_2, \\
Y^{\prime }_{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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0.0... |
dc88af3d5cd6139bbbcc33970b98f02d76b777d1 | subsection | 45 | 160 | Interpolating estimation problem | When the (t,\epsilon )-dependent observations (REF ) are considered, it reads_{t,\epsilon }(,_1, ;_{t,\epsilon },_{t,\epsilon }^{^{\prime }},{1},{2},)
- \sum _{\mu =1}^{n_2} \ln P_{\rm out,2} ( Y_{t,\epsilon ,\mu } |s_{t, \epsilon ,\mu })\\
+ \frac{1}{2} \sum _{i=1}^{n_1}\Bigg [
\sqrt{R_1(t,\epsilon )}
\bigg ( \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 | [
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0b309009800b4a1a44225301fa8fdf6ce8c1e144 | subsection | 46 | 160 | Interpolating estimation problem | \mathbf {n} and the choice of (q_{\epsilon })_{\epsilon \in {\cal B}_{n_0}}, (r_{\epsilon })_{\epsilon \in {\cal B}_{n_0}}.
A simple computation shows that \frac{\partial f_{\mathbf {n},\epsilon }(0)}{\partial \epsilon _1} = - \frac{n_1}{n_0} \langle \mathcal {L} \rangle _{\mathbf {n},0,\epsilon }, where\mathcal {L} ... | {
"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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... |
31cd3409f7fac5731ab47601a2007d2c09c00236 | subsection | 47 | 160 | Interpolating estimation problem | In a similar fashion to what is done in Appendix REF , we compute\Big \vert \frac{\partial f_{\mathbf {n},\epsilon }(0)}{\partial {\epsilon _2}}\Big \vert = \frac{1}{2 n_0}\sum _{\mu =1}^{n_2}
\big \vert \big [u^{\prime }_{Y_{0,\epsilon ,\mu }}(S_{0,\epsilon ,\mu })\langle u^{\prime }_{Y_{0,\epsilon ,\mu }}(s_{0,\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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a7e04dccb64c38f12305cbd8bb47ecf234c3d711 | subsection | 48 | 160 | Interpolating free entropy at t=0 and t=1 | We will denote by _{n_0}(1) any quantity that vanishes uniformly in t \in [0,1] and \epsilon when n_0 \rightarrow +\infty .
It is easily shown (the first equality uses Lemma REF ) that\left\lbrace
\begin{array}{lll}
f_{\mathbf {n},\epsilon }(0)
&=& f_{\mathbf {n},\epsilon =(0,0)}(0) + _{n_0}(1)
= f_{\mathbf {n}} - \fr... | {
"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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20b434027ba632792265af0fed446b66e799d8f2 | subsection | 49 | 160 | Interpolating free entropy at t=0 and t=1 | Applying Theorem 1 of , then (REF ), the free entropy \tilde{f}_{(n_0,n_1),\epsilon } in the thermodynamic limit n_0, n_1 \rightarrow +\infty such that {n_1}{n_0} \rightarrow \alpha _1 satisfies\tilde{f}_{n_0,n_1}
&= _{n_0}(1) +
{\sup }_{q_0 \in [0,\rho _0]} {\inf }_{r_0 \ge 0} \Big \lbrace \psi _{P_0}(r_0) + \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 | [
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5c46385bfc1655257463024fd3191df77c9f068f | subsection | 50 | 160 | Interpolating free entropy at t=0 and t=1 | The second line follows from the Lipschitzianity in both its arguments of the continuous mapping (q,\rho ) \mapsto \Psi _{P_{\rm out},2}\big (q;\rho \big ) on the compact \big \lbrace (q,\rho ): 0 \le \rho \le 1 + \rho _{\mathrm {max}}, 0 \le q \le \rho \big \rbrace , with \rho _{\mathrm {max}} a upper bound on the seq... | {
"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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52782e4ad9b7f023f80c6de01fcdba5ab4146d20 | subsection | 51 | 160 | Interpolating free entropy at t=0 and t=1 | Reporting (REF ) and (REF ) in (REF ), we obtain:f_{\mathbf {n},\epsilon }(1) = -\frac{\alpha _1}{2}\bigg (1 + \rho _1 \int _0^1 \!\! r_{\epsilon }(t) dt\bigg )
+ \alpha \Psi _{P_{\rm out, 2}}\bigg (\int _0^1 q_{\epsilon }(t) dt;\rho _1(n_0)\bigg )\\
+ {\sup }_{q_0 \in [0,\rho _0]} {\inf }_{r_0 \ge 0} \; \bigg \lbrace ... | {
"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... |
7d3086f3237c55ba5c9e9216616d142061493ec9 | subsection | 52 | 160 | Free entropy variation along the interpolation path | From the Fundamental Theorem of Calculus and (\ref {eq:f0_f1}), (\ref {eq:f1_thermoLimit})f_{\mathbf {n}} &= f_{\mathbf {n},\epsilon }(0) + \frac{1}{2}\frac{n_1}{n_0} + _{n_0}(1)\\
&= f_{\mathbf {n},\epsilon }(1) - \int _0^1\frac{df_{\mathbf {n},\epsilon }(t)}{dt} dt
+ \frac{1}{2}\alpha _1 + _{n_0}(1)\\
&= -\frac{\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 | [
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d8602c71f33fdaaf3df9ef74bf1d1bc6ffb47e68 | subsection | 53 | 160 | Free entropy variation along the interpolation path | The overlap \widehat{Q}:
\widehat{Q} \frac{1}{n_1}\sum _{i=1}^{n_1}
\varphi _1\bigg (\bigg [\frac{{1} }{\sqrt{n_0}}\bigg ]_i,_{1,i}\bigg ) \varphi _1\bigg (\bigg [\frac{{1} ^0}{\sqrt{n_0}}\bigg ]_i, _{1,i}\bigg )\,.In Appendix we show
[Free entropy variation]
The derivative of the free entropy (REF ) verifies, for al... | {
"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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b0e6388f6a017257252c67e933262c10ef3f904a | subsection | 54 | 160 | Overlap concentration | An important quantity appearing naturally in the t-derivative of the average free entropy is \widehat{Q}, the overlap between the hidden output ^1 and the sample
^1 \varphi _{1}\big ({{1} }{\sqrt{n_0}}, _{1}\big ) where the triplet (,,_1) is sampled from the posterior distribution associated to the Gibbs bracket \langl... | {
"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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f3f8c24a36a5605d6363e2e4e86c465ceab632b9 | subsection | 55 | 160 | Overlap concentration | For now we assume that we can take q_\epsilon (t) = \big [\langle \widehat{Q} \rangle _{\mathbf {n},t,\epsilon }\big ] and prove
Assume that REF , REF , REF hold, that the interpolation functions (q_{\epsilon })_{\epsilon \in {\cal B}_{n_0}}, (r_{\epsilon })_{\epsilon \in {\cal B}_{n_0}} are regular, and that \forall... | {
"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... |
eecbbcbc4358c971caf9549127ab124ebb12582d | subsection | 56 | 160 | Overlap concentration | 2}\\
\le \frac{1}{s_{n_0}^2} \int _{{\cal B}_{n_0}} \!\!\!\! d\epsilon \int _0^1 \!\!dt\,
\Bigg [\Bigg \langle \Bigg (\frac{1}{n_1}\sum _{\mu =1}^{n_2} u_{Y_{t,\epsilon ,\mu }}^{\prime }( S_{t,\epsilon ,\mu } )u_{Y_{t,\epsilon ,\mu }}^{\prime }(s_{t,\epsilon ,\mu }) - r_{\epsilon }(t)\Bigg )^{\! 2}
\Bigg \rangle _{\! \... | {
"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",
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] | 2,018 | en | Computer Science | [
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0.... |
1e5b2a51dfb7fde9076beb7ab0c72f06dee822c7 | subsection | 57 | 160 | Overlap concentration | 2}
\Bigg \rangle _{\! \mathbf {n},t,\epsilon }\Bigg ]\\
&\qquad \qquad \qquad \qquad \qquad \qquad \le \frac{1}{n_1^2}\Big [
\Big \Vert \big \lbrace u_{Y_{t,\epsilon ,\mu }}^{\prime }( S_{t,\epsilon ,\mu })\big \rbrace _{\mu =1}^{n_2} \Big \Vert \cdot \Big \langle \Big \Vert \big \lbrace u_{Y_{t,\epsilon ,\mu }}^{\prim... | {
"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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... |
9ace7ff9c3bed9142fe27fda8b9e37d3ba85d116 | subsection | 58 | 160 | Overlap concentration | \mathbf {n},t,\epsilon }\Big ]\\
&\qquad \qquad \qquad \qquad \qquad \qquad \le \frac{1}{n_1^2}\Big [
\Big \Vert \big \lbrace u_{Y_{t,\epsilon ,\mu }}^{\prime }(S_{t,\epsilon ,\mu })\big \rbrace _{\mu =1}^{n_2}\Big \Vert ^2
\Big ]
= \frac{n_2}{n_1^2}\Big [ u_{Y_{t,\epsilon ,1}}^{\prime }(S_{t,\epsilon ,1})^2\Big ]\,.Th... | {
"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.... |
683a193df55fca2356b1f0755ad2430c5aac0862 | subsection | 59 | 160 | Overlap concentration | Putting everything together, we obtain the following bound uniform w.r.t. the choice of the interpolating functions:\frac{1}{s_{n_0}^2} \int _{{\cal B}_{n_0}} \!\!\!\! d\epsilon \int _0^1 \!\!dt\,
\Bigg [\Bigg \langle \Bigg (\frac{1}{n_1}\sum _{\mu =1}^{n_2} u_{Y_{t,\epsilon ,\mu }}^{\prime }( S_{t,\epsilon ,\mu } )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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0.006378862075507641,
0.03506848216056824,
... |
02738cdd6bb4fb8a33fbd14021dfab1cbd1df1d2 | subsection | 60 | 160 | Overlap concentration | \mathbf {n},t,\epsilon }\,\Bigg ]\Bigg \vert \\
\le \frac{C(\varphi _1,\varphi _2,\alpha _1,\alpha _2, S)}{n^{{1}{16}}}\,.Therefore the integral of (REF ) over (t,\epsilon ) reads\frac{1}{s_{n_0}^2}\int _{{\cal B}_{n_0}} \!\!\!\! d\epsilon \int _0^1 \!\! dt\, \frac{df_{\mathbf {n},\epsilon }(t)}{dt}
&= \frac{1}{s_{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 | [
0.009252323769032955,
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0.007757306564599276,
-0.01125076413154602,
... |
5080d33024c51340d2fc5895ce6d084eca3b02ed | subsection | 61 | 160 | Lower and upper matching bounds | To end the proof of Theorem REF one has to go through the following two steps:Prove that under the assumptions REF , REF and REF
\lim _{\mathbf {n} \rightarrow \infty }f_{\mathbf {n}}
= {\sup }_{r_1 \ge 0} {\inf }_{q_1 \in [0,\rho _1]} {\sup }_{q_0 \in [0,\rho _0]} {\inf }_{r_0 \ge 0} f_{\rm RS}(q_0,r_0,q_1,r_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 | [
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0... |
945e7b482c0484529ec92644fc22c92f3124ee26 | subsection | 62 | 160 | Lower and upper matching bounds | Fix (t,r_1,r_2) \in D_{\mathbf {n}}^{\circ }.
Consider the two sets of observations{\left\lbrace \begin{array}{ll}
Y_{t,r_2,\mu } &\sim P_{\rm out, 2}(\ \cdot \ | \, S_{t,r_2,\mu })\,, \qquad \qquad \;\;\; 1 \le \mu \le n_2, \\
Y^{\prime }_{t,r_1,i} &= \sqrt{r_1}\, \varphi _1\Big (\Big [\frac{{1} ^0}{\sqrt{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 | [
-0.0012837464455515146,
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-0.01698436588048935,
-0.001260856399312615... |
03f519020095b426dc144e4ebaad172666c26a10 | subsection | 63 | 160 | Lower and upper matching bounds | The Gibbs bracket corresponding to this Hamiltonian is denoted \langle - \rangle _{\mathbf {n},t,r_1,r_2} and is used to define F_{\mathbf {n}}:F_{\mathbf {n}}(t,r_1,r_2)
\big [\langle \widehat{Q} \rangle _{\mathbf {n},t,r_1,r_2}\big ]
= \frac{1}{n_1} \sum _{i=1}^{n_1} \big [X_i^1 \langle x_i^1 \rangle _{\mathbf {n},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 | [
-0.0363384373486042,
-0.007887057960033417,
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0.010373693890869617,
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-0.0118382154032588,
0... |
f11dfc2cbc2149d33c90e019bff99322b1392df2 | subsection | 64 | 160 | Lower and upper matching bounds | Then \mathrm {MMSE}\big ( ^1 \, \big \vert \, _{t,r_2}, \, _{t,r_1}^{\prime },{1},{2},\big ) is clearly a non-increasing function of r_1, so F_{\mathbf {n}}(t,r_1,r_2) is a non-decreasing function of r_1.r_2's only role is in the generation process of _{t,r_2}:_{t,r_2} \sim P_{\rm out}\Big ( \cdot \, \Big \vert \, \sqr... | {
"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 | [
-0.024056674912571907,
0.0006464182515628636,
-0.007146830670535564,
0.021204045042395592,
0.009572329930961132,
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0.024514315649867058,
-0.025841476395726204,
-0.03148571774363518,
-0.03298068046569824,
... |
b7212eb52a7b8fc5220792949f5b70577a9a3444 | subsection | 65 | 160 | Lower and upper matching bounds | Now notice that\mathrm {MMSE}\big ( ^1 \, \big \vert \, _{t,r_2}, \, _{t,r_1}^{\prime },{1},{2},\big )
&\ge \mathrm {MMSE}\big ( ^1 \, \big \vert \, \widetilde{}_{t,r_2,r_2^{\prime }}, _{t,r_1}^{\prime },{1},{2},\big )\,;\\
\mathrm {MMSE}\big ( ^1 \, \big \vert \, _{t,r_2^{\prime }}, \, _{t,r_1}^{\prime },{1},{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 | [
-0.04953405261039734,
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0... |
1b52760847ce5bd923f1a9d0451328aa57a1465e | subsection | 66 | 160 | Lower and upper matching bounds | Define for t \in [0,1]r_{\epsilon }(t) R_1^{\prime }(t,\epsilon ) = r
\qquad \text{and} \qquad 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,\epsilon ) = \epsilon _1 + \int _0^t r_\epsilon (s) ds and R_2(t,\epsilon ) = \epsilon _2 + \int _0^t q_\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 | [
-0.03415504842996597,
0.046913594007492065,
-0.04798189178109169,
-0.0014021425740793347,
0.03598641976714134,
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0.006585300900042057,
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... |
5d2b004930dd408027ca6597b998578191ddedf7 | subsection | 67 | 160 | Lower and upper matching bounds | Moreover, since also the Jacobian determinant 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 can now be applied to this special choice of regular functions:f_{\mathbf {n}}
&= _{n_0}(1) + \frac{1}{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 | [
-0.053343117237091064,
0.02401660941541195,
-0.01278647966682911,
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0.022856976836919785,
-0.026961466297507286,
0... |
a714e7882f1c2fdbf5b3488d821d87c0fc49e8a3 | subsection | 68 | 160 | Lower and upper matching bounds | Define on [0,+\infty [ \times [0,\rho _1] the function \psi (r_1, q_1)f(r_1) + g(q_1) - \frac{\alpha _1}{2} r_1 q_1 where f: [0,+\infty [ \rightarrow \mathbb {R} and g: [0,\rho _1] \rightarrow \mathbb {R} are such thatf(r_1) \!\!{\sup }_{q_0 \in [0,\rho _0]} {\inf }_{r_0 \ge 0} \Big \lbrace \psi _{P_0}(r_0) + \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 | [
-0.016009269282221794,
0.027653761208057404,
-0.009729179553687572,
-0.007726113311946392,
0.011522401124238968,
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0.04209110140800476,
-0.008531155064702034,
-0.012018398381769657,
-0.03610860928893089,
0... |
2315b666ca61b2ba881d2aa70bb8ad2d03d51a10 | subsection | 69 | 160 | Lower and upper matching bounds | Hence\forall r_1 \ge : \inf _{q_1 \ge [0,\rho _1]} \psi (r_1, q_1) = \psi (r_1, \rho _1) \,.The latter implies that for all r_1 \ge we have\inf _{q_1 \ge [0,\rho _1]} \psi (r_1, q_1) - \inf _{q_1 \ge [0,\rho _1]} \psi (, q_1)
&= \psi (r_1, \rho _1) - \psi (, \rho _1)\\
&= f(r_1) - f() - \frac{\alpha _1 \rho _1}{2}(r_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 | [
-0.007677751127630472,
0.04625720903277397,
0.009680145420134068,
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0.0143180713057518,
-0.009954758919775486,
-0.02952100895345211,
-0.046623360365629196,
0.013... |
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