measuring and exploiting data-model fit with sheaves · 5/24/2018 · michael robinson key points...
TRANSCRIPT
![Page 1: Measuring and Exploiting Data-Model Fit with Sheaves · 5/24/2018 · Michael Robinson Key points Systems can be encoded as sheaves Datasets are assignments to a sheaf model of a](https://reader034.vdocuments.mx/reader034/viewer/2022050206/5f595440b6632a3ff05627ab/html5/thumbnails/1.jpg)
Michael Robinson
Measuring and Exploiting Data-Model Fit
with Sheaves
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Michael Robinson
Acknowledgements● Collaborators:
– Cliff Joslyn, Katy Nowak, Brenda Praggastis, Emilie Purvine (PNNL)– Chris Capraro, Grant Clarke, Janelle Henrich (SRC)– Jason Summers, Charlie Gaumond (ARiA)– J Smart, Dave Bridgeland (Georgetown)
● Students:– Eyerusalem Abebe– Olivia Chen– Philip Coyle– Brian DiZio
● Recent funding: DARPA, ONR, AFRL
– Jen Dumiak– Samara Fantie– Sean Fennell– Robby Green
– Noah Kim– Fangfei Lan– Metin Toksoz-Exley– Jackson Williams– Greg Young
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Michael Robinson
The middle way
Statistics
Topology
Geometry
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Michael Robinson
Key points● Systems can be encoded as sheaves● Datasets are assignments to a sheaf model of a system● Consistency radius measures compatibility between
system and dataset● Statistical inference minimizes consistency radius
– Regression (parameter estimation, learning, model selection) varies the sheaf
– Data fusion (prediction) varies the assignment● Topological properties of the minimization problem
may aid model selection and/or data cleaning
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Michael Robinson
Process flow diagramSystem model
Sheaf
Observations
Assignment
Consistency radiusPaired (Sheaf,Assignment)
Regression
Data fusion
“Syntax”
“Semantics”
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Michael Robinson
CategoriallySystem model
Shv
Observations
Pseud
Consistency radiusShvFPA
Regression
Data fusion
“Syntax”
“Semantics”
(Faithful)
(Faithful)
(Forgetful)
(Forgetful)
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Michael Robinson
Geometrically/topologicallySystem model
Shv
Observations
Pseud
ℝShvFPA
Regression
Data fusion
“Syntax”
“Semantics”
(Faithful)
(Faithful)
(Continuous)
(Continuous)
(Continuous)
(Note: all of this within an isomorphism class of sheaves)
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Michael Robinson
What is a sheaf?
A sheaf of _____________ on a ______________(data type) (topological space)
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Michael Robinson
What is a sheaf?
A sheaf of _____________ on a ______________(data type) (topological space)
pseudometric spaces partial order
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Michael Robinson
Topologizing a partial order
Open sets are unionsof up-sets
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Michael Robinson
Topologizing a partial order
Intersectionsof up-sets are alsoup-sets
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Michael Robinson
( )( )
A sheaf on a poset is...
A set assigned to each element, calleda stalk, and …
ℝ
ℝ2 ℝ2
ℝ3
0 1 11 0 1
1 0 10 1 1
(1 0)(0 1)
ℝ ℝ2
ℝ2
(1 -1) ( )0 11 0
( )-3 3-4 4
This is a sheaf of vector spaces on a partial order
ℝ3
( )0 1 11 0 1
ℝ
(-2 1)
(The stalk on an element in the posetis better thought of beingassociated to the up-set)
231
2 -23 -31 -1
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Michael Robinson
( )( )
A sheaf on a poset is...
… restriction functions between stalks, following the order relation…
ℝ
ℝ2 ℝ2
ℝ3
0 1 11 0 1
1 0 10 1 1
(1 0)(0 1)
ℝ ℝ2
ℝ2
(1 -1)
231
( )0 11 0
( )-3 3-4 4
2 -23 -31 -1
This is a sheaf of vector spaces on a partial order
ℝ3
( )0 1 11 0 1
ℝ
(-2 1)
(“Restriction” because it goes frombigger up-sets to smaller ones)
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Michael Robinson
( )( )
A sheaf on a poset is...ℝ
ℝ2 ℝ2
ℝ3
0 1 11 0 1
1 0 10 1 1
(1 0)(0 1)
ℝ ℝ2
ℝ2
(1 -1) ( )0 11 0
( )-3 3-4 4
This is a sheaf of vector spaces on a partial order
ℝ3
( )0 1 11 0 1
ℝ
(-2 1)
(1 -1) =
( )0 1 11 0 1(1 0) = (0 1) ( )1 0 1
0 1 1
( )0 11 0( )-3 3
-4 4( )1 0 10 1 1
2 -23 -31 -1
=
… so that the diagramcommutes!
231
231
2 -23 -31 -1
2 -23 -31 -1
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Michael Robinson
( )( )
An assignment is...
… the selection of avalue on some open sets
0 1 11 0 1
1 0 10 1 1
(1 0)(0 1)
(1 -1)
231
( )0 11 0
( )-3 3-4 4
2 -23 -31 -1 ( )0 1 1
1 0 1
(-2 1)
(-1)
( )23
( )-4-3
( )-3-4
( )32
(-4)
(-4)
230
-2-3-1
The term serration is more common, but perhaps more opaque.
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Michael Robinson
( )( )
A global section is...
… an assignmentthat is consistent with the restrictions
0 1 11 0 1
1 0 10 1 1
(1 0)(0 1)
(1 -1)
231
( )0 11 0
( )-3 3-4 4
2 -23 -31 -1 ( )0 1 1
1 0 1
(-2 1)
(-1)
( )23
( )-4-3
( )-3-4
( )32
(-4)
(-4)
230
-2-3-1
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Michael Robinson
( )( )
Some assignments aren’t consistent
… but they mightbe partially consistent
0 1 11 0 1
1 0 10 1 1
(1 0)(0 1)
(1 -1)
231
( )0 11 0
( )-3 3-4 4
2 -23 -31 -1 ( )0 1 1
1 0 1
(-2 1)
(+1)
( )23
( )-4-3
( )-3-4
( )32
(-4)
(-4)
231
-2-3-1
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Michael Robinson
( )( )
Consistency radius is...… the maximum (or some other norm)distance between the value in a stalk and the values propagated along the restrictions
0 1 11 0 1
1 0 10 1 1
(1 0)(0 1)
(1 -1)
231
( )0 11 0
( )-3 3-4 4
2 -23 -31 -1 ( )0 1 1
1 0 1
(-2 1)
(+1)
( )23
( )-4-3
( )-3-4
( )32
(-4)
(-4)
231
-2-3-1
231( )0 1 1
1 0 1 ( )32- = 2
( )23(1 -1) - 1 = 2
(+1) - 231
-2-3 = 2 14-1 MAX ≥ 2 14
Note: lots more restrictions to check!
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Michael Robinson
( )( )
Consistency radius is monotonic
Lemma: Consistency radius on an open set U is computed by only considering open sets V1 ⊆ V2 ⊆ U
0 1 11 0 1
1 0 10 1 1
(1 0)(0 1)
(1 -1)
231
( )0 11 0
( )-3 3-4 4
2 -23 -31 -1 ( )0 1 1
1 0 1
(-2 1)
(+1)
( )23
( )-4-3
( )-3-4
( )32
(-4)
(-4)
231
-2-3-1
Note: lots more restrictions to check!
ConsistencyRadius = 0
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Michael Robinson
( )( )
Consistency radius is monotonicProposition: If U ⊆ V then c(U) ≤ c(V)
0 1 11 0 1
1 0 10 1 1
(1 0)(0 1)
(1 -1)
231
( )0 11 0
( )-3 3-4 4
2 -23 -31 -1 ( )0 1 1
1 0 1
(-2 1)
(+1)
( )23
( )-4-3
( )-3-4
( )32
(-4)
(-4)
231
-2-3-1
Note: lots more restrictions to check!
ConsistencyRadius = 2 14
Consistencyradius restricted toan open set U
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Michael Robinson
Consistency radius optimization
ℝ ℝ
ℝ
ℝ(2)
(1)
(1)
(1)
(½)
This is a sheaf on a small poset
A B
C
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Michael Robinson
Consistency radius optimization
0 1
?
?(2)
(1)
(1)
(1)
(½)
Here is an assignment supported on part of it
A B
C
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Michael Robinson
Consistency radius optimization
0 1
?(2)
(1)
(1)
(1)
(½)½
Minimizing the consistency radius when extending globally
A B
C
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Michael Robinson
Consistency radius optimization
⅓
?(2)
(1)
(1)
(1)
(½)⅓
Here is the closest global section (everything can be changed)
⅔A B
C
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Michael Robinson
Consistency radius is not a measure
ℝ ℝ
ℝ
ℝ(2)
(1)
(1)
(1)
(½)
This is the full sheaf diagram including all open sets in theAlexandrov topology, not just the base
{A,C} {B,C}
{C}
{A,B,C}
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Michael Robinson
Consistency radius is not a measure
0 1
?
?(2)
(1)
(1)
(1)
(½)
{A,C} {B,C}
{C}
{A,B,C}
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Michael Robinson
Consistency radius is not a measure
0 1
½
⅓(2)
(1)
(1)
(1)
(½)
Minimizing the consistency radius when extendingThe value on the intersection is no longer unique!
This value can be anything between ⅓ and ⅔
{A,C} {B,C}
{C}
{A,B,C}
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Michael Robinson
Consistency radius is not a measure
0 1
½
⅓(2)
(1)
(1)
(1)
(½)
Consistency radius of this open set = 0
Lemma: Consistency radius on an open set U is computed by only considering open sets V1 ⊆ V2 ⊆ U
{A,C} {B,C}
{C}
{A,B,C}
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Michael Robinson
Consistency radius is not a measure
0 1
½
⅓(2)
(1)
(1)
(1)
(½)
c(U) = ½
Lemma: Consistency radius on an open set U is computed by only considering open sets V1 ⊆ V2 ⊆ U
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Michael Robinson
Consistency radius is not a measure
0 1
½
⅓(2)
(1)
(1)
(1)
(½)
c(U) = ½ c(V) = ½
Lemma: Consistency radius on an open set U is computed by only considering open sets V1 ⊆ V2 ⊆ U
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Michael Robinson
Consistency radius is not a measure
0 1
½
⅓(2)
(1)
(1)
(1)
(½)
c(U) = ½ c(V) = ½
c(U ∩ V) = 0
c(U ∪ V) = ⅔ ≠ c(U) + c(V) – c(U ∩ V)
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Michael Robinson
The consistency filtration
1
⅓
(1)
(1)
Consistencythreshold 0 ½ ⅔
0 1
½
⅓(2)
(1)
(1)
(1)
(½)
0 1
½
⅓(2)
(1)
(1)
(1)
(½)
0 1
½
⅓(2)
(1)
(1)
(1)
(½)
Consistency radius = ⅔
● … assigns the set of open sets with consistency less than a given threshold
● Lemma: consistency filtration is a sheaf of collections of open sets on (ℝ,≤). Restrictions in this sheaf are coarsenings.
coarsen coarsen
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Michael Robinson
The consistency filtration
1
⅓
(1)
(1)
0 ½ ⅔
0 1
½
⅓(2)
(1)
(1)
(1)
(½)
0 1
½
⅓(2)
(1)
(1)
(1)
(½)
0 1
½
⅓(2)
(1)
(1)
(1)
(½)
Consistency radius = ⅔
● Theorem: Consistency filtration is a functor
CF : ShvFPA→ Shv(ℝ,≤)● As a practical matter, CF transforms an assignment into
persistent Čech cohomology
Consistencythreshold
coarsen coarsen
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Michael Robinson
The consistency filtration
1
⅓
(1)
(1)
0 ½ ⅔
0 1
½
⅓(2)
(1)
(1)
(1)
(½)
0 1
½
⅓(2)
(1)
(1)
(1)
(½)
0 1
½
⅓(2)
(1)
(1)
(1)
(½)
Consistency radius = ⅔
● Theorem: Consistency filtration is continuous under an appropriate interleaving distance, so– Generators of PH0(CF) with are highly consistent subsets– Generators of PHk(CF) for k > 0 are robust obstructions to
consistency
Consistencythreshold
coarsen coarsen
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Michael Robinson
Geometrically/topologicallySystem model
Shv
Observations
Pseud
ℝShvFPA
Regression
Data fusion
“Syntax”
“Semantics”
(Faithful)
(Faithful)
(Continuous)
(Continuous)
(Continuous)
(Note: all of this within an isomorphism class of sheaves)
![Page 36: Measuring and Exploiting Data-Model Fit with Sheaves · 5/24/2018 · Michael Robinson Key points Systems can be encoded as sheaves Datasets are assignments to a sheaf model of a](https://reader034.vdocuments.mx/reader034/viewer/2022050206/5f595440b6632a3ff05627ab/html5/thumbnails/36.jpg)
Michael Robinson
Inference and consistencyStatistical inference minimizes various risks:
● Intrinsic risk– Noise in the data
● Parameter risk– Errors in estimation
● Model risk– Using the wrong
model
Minimize consistency radius by:
Varying the assignment
Varying the sheaf isomorphism class
Varying the sheaf within an isomorphism class
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Michael Robinson
Chris Capraro, Janelle Henrich
Separating sinusoids from noise● Consider a signal formed from N sinusoids
– Each sinusoid has a (real) frequency ω– Each sinusoid has a (complex) amplitude a
● Task: Recover these parameters from M samples f is an arbitrary
known functionPossibilities:● Magnitude● Phase● Identity function● Quantizer output● Signal dispersion
Gaussian noiseSample time
Sinusoid frequency
Sinusoid amplitude
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Michael Robinson
Chris Capraro, Janelle Henrich
Separating sinusoids from noise● Consider a signal formed from N sinusoids
– Each sinusoid has a (real) frequency ω– Each sinusoid has a (complex) amplitude a
● Model the situation as a sheaf over a poset...
ℂ ℂ ℂ…
Parameter spaces become stalks
Signal models become restrictions
ℝN×ℂN
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Michael Robinson
Chris Capraro, Janelle Henrich
Separating sinusoids from noise● Consider a signal formed from N sinusoids
– Each sinusoid has a (real) frequency ω– Each sinusoid has a (complex) amplitude a
● The samples become an assignment to part of the sheaf
ℝN×ℂN
x1 ∈ ℂ x2 ∈ ℂ xM ∈ ℂ Observed…
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Michael Robinson
Chris Capraro, Janelle Henrich
Separating sinusoids from noise● Consider a signal formed from N sinusoids
– Each sinusoid has a (real) frequency ω– Each sinusoid has a (complex) amplitude a
● Find the unknown parameters by minimizing consistency radius
(ω1,…,ωN,a1,…,aN) ∈ ℝN×ℂN
x1 ∈ ℂ x2 ∈ ℂ xM ∈ ℂ Observed
Inferred
…
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Michael Robinson
Chris Capraro, Janelle Henrich
Sheaves deliver excellent performance
8x improvement over state-of-the-art in heavy noise!
Sheaf result
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Michael Robinson
The future● Preprint: https://arxiv.org/abs/1805.08927● Computational sheaf theory
– Small examples can be put together ad hoc– Larger ones require a software library
● PySheaf: a software library for sheaves– https://github.com/kb1dds/pysheaf– Includes several examples you can play with!
● Connections to statistical models need to be explored– Is consistency radius related to likelihood functions?
● Extensive testing on various datasets and scenarios