The Geometry of the Identi ability of Tensors · 2020-01-03 · Luca Chiantini (Universita’ di...

The Geometry of the Identifiability of Tensors

Luca Chiantini

Universita’ di Siena - Italy

Nov 10 - 2014

Luca Chiantini (Universita’ di Siena - Italy) Identifiability of Tensors Nov 10 - 2014 1 / 38

Index of the talk

1 Introduction

2 The Geometry of Tensors

3 Local analysis

4 Generic identifiability

5 Specific tensors

Tensors

Tensor of dimension n = n-dimensional matrix.

T = (Ti1,...,in)

Why multi-dimensional matrices?

Matrices are powerful tools for representing relations between two sets.

When one needs to relate three or more sets, then higher-dimensionaltensors enter into the picture.

Tensors

T = (Ti1,...,in)

Tensors

T = (Ti1,...,in)

Simple tensors

Definition

A tensor T = (Ti1,...,in) is simple when there are vectors v1, . . . , vn suchthat

Ti1,...,in = v1i1 · · · vninfor all choices of the indexes.

Notation: T = v1 ⊗ · · · ⊗ vn.

If the tensor T represents some statistical connection among variables,then T is simple exactly when the variables are independent.

Simple tensors

The set of simple tensors is a cone, but not a subspace.It spans the whole space of tensors.

Simple tensors

Definition

Simple tensors

Definition

Simple tensors

Definition

Simple tensors

The set of simple tensors is a cone, but not a subspace.

It spans the whole space of tensors.

Simple tensors

Definition

Simple tensors

The complexity of a tensor

The cone of simple tensors spans the whole space of tensors.

Thus, any tensor is a sum of simple ones.

Definition

The rank of a tensor T is the minimum r such that

T =r∑

where all the Ti ’s are simple.∑ri=1 Ti is a decomposition of T .

The rank is a good measure for the complexity of T .In the statistical interpretation, tensors of high rank imply manyconnections among the variables.

Definition

T =r∑

Definition

T =r∑

where all the Ti ’s are simple.

∑ri=1 Ti is a decomposition of T .

Definition

T =r∑

Definition

T =r∑

The rank is a good measure for the complexity of T .

In the statistical interpretation, tensors of high rank imply manyconnections among the variables.

Definition

T =r∑

The geometric point of view

Algebraic Geometry of tensors

Many interesting sets of tensors are defined by algebraic equations (outsidea set of measure 0).

Thus, following the point of view of AlgebraicGeometry, it is natural to start with coefficients in an algebraically closedfield as the complex field C.

Tensors over real numbers are considered at a second level of difficulty.

Projective tensors

Many interesting sets of tensors are cones (i.e. defined by homogeneouspolynomials). Thus, it is natural to factor out the C∗-action.That is: we consider tensors as points in a projective space PN .

The projectification of tensors is the analogue of the normalization, in thestatistical approach.

Many interesting sets of tensors are defined by algebraic equations (outsidea set of measure 0). Thus, following the point of view of AlgebraicGeometry, it is natural to start with coefficients in an algebraically closedfield as the complex field C.

Projective tensors

Many interesting sets of tensors are cones

(i.e. defined by homogeneouspolynomials). Thus, it is natural to factor out the C∗-action.That is: we consider tensors as points in a projective space PN .

Projective tensors

Many interesting sets of tensors are cones (i.e. defined by homogeneouspolynomials).

Thus, it is natural to factor out the C∗-action.That is: we consider tensors as points in a projective space PN .

Projective tensors

Many interesting sets of tensors are cones (i.e. defined by homogeneouspolynomials). Thus, it is natural to factor out the C∗-action.

That is: we consider tensors as points in a projective space PN .

Projective tensors

Problems on the Geometry of tensors

One would understand the behavior of the algebraic varieties of tensorsof given rank. For instance:

Generic tensors of given rank

find the dimension of the variety;

find (implicit) equations for the variety;

find how many minimal decompositions a general tensor of given rankhas (identifiability problem).

Specific tensors

compute the rank of a given tensor;

find how many minimal decompositions a given tensor has(identifiability problem);

find the decompositions of a given tensor.

Specific tensors

find how many minimal decompositions a general tensor of given rankhas

(identifiability problem).

Specific tensors

find how many minimal decompositions a given tensor has

(identifiability problem);

Specific tensors

Projective Geometry: secant varieties

Let X be any projective variety in PN .

The abstract k-th secant variety of X is the subset Aσk ⊂ X k × PN of(k + 1)-tuples (P1, . . . ,Pk ,Q) where P1, . . . ,Pk are independent pointsin X and Q belongs to the linear span of P1, . . . ,Pk .

The obvious projection ak : Aσk → PN determines:

The strict k-th secant variety σ0k of a variety X = the image of ak .

I.e.: σ0k = {Q : there are independent pts P1, . . . ,Pk ∈ X with

P ∈ 〈P1, . . . ,Pk〉}.The k-th secant variety σk = the (Zariski) closure of the strict k-secantvariety.

P ∈ 〈P1, . . . ,Pk〉}.

The k-th secant variety σk = the (Zariski) closure of the strict k-secantvariety.

Picture

The k-th secant variety σk is the (Zariski) closure of the strict k-secantvariety.

Geometric interpretation

Simple tensors of type n1 × · · · × ns

⇐⇒ tensors which are products T = v1 ⊗ · · · ⊗ vn, vi ∈ Cni

⇐⇒ points of the Segre product Pn1−1 × · · · × Pns−1 which is not alinear space but can be embedded in a projective space PN .The image is the Segre embedding X of Pn1−1 × · · · × Pns−1

rank 1 tensors i.e. simpletensors

⇐⇒ points of the Segre embedding X ofPn1−1 × · · · × Pns−1

rank k tensors ⇐⇒ points of the strict k-th secantvariety of the Segre embedding X .

tensors of border rank k (i.e.limits of rank k tensors)

⇐⇒ points of the k-th secant variety ofthe Segre embedding X .

Simple tensors of type n1 × · · · × ns⇐⇒ tensors which are products T = v1 ⊗ · · · ⊗ vn, vi ∈ Cni

⇐⇒ points of the Segre product Pn1−1 × · · · × Pns−1

which is not alinear space but can be embedded in a projective space PN .The image is the Segre embedding X of Pn1−1 × · · · × Pns−1

⇐⇒ points of the Segre product Pn1−1 × · · · × Pns−1 which is not alinear space

but can be embedded in a projective space PN .The image is the Segre embedding X of Pn1−1 × · · · × Pns−1

⇐⇒ points of the Segre product Pn1−1 × · · · × Pns−1 which is not alinear space but can be embedded in a projective space PN .

The image is the Segre embedding X of Pn1−1 × · · · × Pns−1

rank k tensors

⇐⇒ points of the strict k-th secantvariety of the Segre embedding X .

tensors of border rank k

(i.e.limits of rank k tensors)

Geometric translation

Find the dimension of thevariety of tensors of (border)rank k

⇐⇒ Find the dimension of secantvarieties of Segre varieties

Find equations for the varietyof tensors of rank k

⇐⇒ Find equations for secantvarieties of Segre varieties

Find the decomposition oftensors of rank k

⇐⇒Find k points of a Segrevariety X which spanP ∈ σk(X ).

Study the identifiability oftensors of rank k

⇐⇒Find how many sets of kpoints of a Segre variety Xspan P ∈ σk(X ).

We will focus on this problem,(mod permutations)both for generic and for specific tensors.

We will focus on this problem,

(mod permutations)both for generic and for specific tensors.

We will focus on this problem,(mod permutations)

both for generic and for specific tensors.

Local method: Terracini’s Lemma

Terracini’s Lemma (1911)

The tangent space at a general point P ∈ σk(X ), P ∈ 〈P1, . . . ,Pk〉, isthe span

〈TX ,P1 , . . . ,TX ,Pk〉

The proof:

Another proof:

Aσk is locally a product X k × Pk−1

and ak is generically smooth.(thus general point = ...)

〈TX ,P1 , . . . ,TX ,Pk〉

The proof:

Another proof:

〈TX ,P1 , . . . ,TX ,Pk〉

The proof:

Another proof:

〈TX ,P1 , . . . ,TX ,Pk〉

The proof:

Another proof:

and ak is generically smooth.

(thus general point = ...)

〈TX ,P1 , . . . ,TX ,Pk〉

The proof:

Another proof:

Consequences of Terracini’s Lemma

Terracini’s Lemma is usually applied as a method for computing thedimension of a secant variety

i.e. a variety of tensors of given rank.

From now on, assume that the dimension of the secant variety σk is the(expected) value k dim(X ) + k − 1 < N in PN (non-defective case).

Terracini’s Lemma can be used also to detect identifiability.

For generic non identifiability:

Terracini’s Lemma is usually applied as a method for computing thedimension of a secant variety i.e. a variety of tensors of given rank.

From now on, assume that the dimension of the secant variety σk is the(expected) value k dim(X ) + k − 1

< N in PN (non-defective case).

From now on, assume that the dimension of the secant variety σk is the(expected) value k dim(X ) + k − 1 < N in PN

(non-defective case).

(contact locus)

Proposition

If the general tensor of a given rank k is not identifiable:then the general space tangent to X at k points P1, . . . ,Pk is indeedtangent at infinitely many points which define a subvariety containingP1, . . . ,Pk : a contact locus Σ.

(contact locus)

Proposition

If the general tensor of a given rank k is not identifiable:

then the general space tangent to X at k points P1, . . . ,Pk is indeedtangent at infinitely many points which define a subvariety containingP1, . . . ,Pk : a contact locus Σ.

(contact locus)

Proposition

If the general tensor of a given rank k is not identifiable:then the general space tangent to X at k points P1, . . . ,Pk is indeedtangent at infinitely many points

which define a subvariety containingP1, . . . ,Pk : a contact locus Σ.

(contact locus)

Proposition

If the general tensor of a given rank k is not identifiable:then the general space tangent to X at k points P1, . . . ,Pk is indeedtangent at infinitely many points which define a subvariety containingP1, . . . ,Pk

: a contact locus Σ.

(contact locus)

Proposition

If the general tensor of a given rank k is not identifiable:then the general space tangent to X at k points P1, . . . ,Pk is indeedtangent at infinitely many points which define a subvariety containingP1, . . . ,Pk : a contact locus Σ.

Degenerate subvarieties

Contact loci have the property that they are contained in manyhyperplanes

(i.e. they are higly degenerate.

Theorem (— Ciliberto, 2006)

Let X ⊂ PN be a variety of dimension n and assume N > nk + k − 1.Then a k-contact locus Σ of dimension m > 0 spans a subspace ofdimension ≤ mk + k − 1.

Generic non-identifiability is strictly related with the existence of (higly)degenerate subvarieties passing through k general points.

Contact loci have the property that they are contained in manyhyperplanes (i.e. they are higly degenerate.

Using degenerate subvarieties

Conversely

by means of degenerate subvarieties (through general points ofa Segre product) one can prove some generic non-identifiabilitystatements.

Theorem (Bocci —, 2011)

General tensors of rank 5 and type 2× 2× 2× 2× 2 (5 times) are notidentifiable. Indeed they have exactly two decompositions.

The proof consists in showing that through 5 general points ofP1×· · ·×P1 (5 times) one can draw an elliptic normal curve of degree 10.

Conversely by means of degenerate subvarieties (through general points ofa Segre product) one can prove some generic non-identifiabilitystatements.

General tensors of rank 5 and type 2× 2× 2× 2× 2 (5 times) are notidentifiable.

Indeed they have exactly two decompositions.

Theorem (— Ottaviani, 2012)

General tensors of rank 6 and type 4× 4× 4 are not identifiable.Indeed they have exactly two decompositions.

Again the proof consists in showing that through 6 general points ofP3 × P3 × P3 one can draw an elliptic normal curve of degree 10.

General tensors of rank 6 and type 4× 4× 4 are not identifiable.

Indeed they have exactly two decompositions.

Special contact loci

The existence of a contact locus does not necessarily imply thenon-identifiability.

It depends on what the contact locus is.

Definition

A variety Y of dimension m in PN , with N = mk + k − 1, is a k-OASS(One Apparent Double Point) if the general point of PN is contained inexactly one k-secant space to Y .

There are varieties which are known to be OASS

rational normal curves;

general arrangements of linear spaces;

OASS surfaces are classified (Ciliberto - Russo (2009)).

Proposition (— Ciliberto, 2006)

If the contact locus is a k-OASS, then X is generically identifiable.

The existence of a contact locus does not necessarily imply thenon-identifiability. It depends on what the contact locus is.

Definition

A variety Y of dimension m in PN , with N = mk + k − 1, is a k-OASS(One Apparent Double Point)

if the general point of PN is contained inexactly one k-secant space to Y .

Definition

Maximal rank for identifiability

The identifiabilty of a generic tensor of rank k and type a1 × · · · × as canbe handled by means of contact loci only if

nk + k − 1 ≤ N

wheren = (

∑ai )− s + 1 = dimension of the Segre variety, i.e. the dimension of

the variety of simple tensors;N = (Πai )− 1 = dimension of the total space of tensors.

This is due to the following easy remark:

ak : Aσk −→ σk ⊂ PN

k-identifiability ⇔ ak is generically finite impossible if dim(Aσk) > N.

nk + k − 1 ≤ N

wheren = (

∑ai )− s + 1 = dimension of the Segre variety,

i.e. the dimension ofthe variety of simple tensors;N = (Πai )− 1 = dimension of the total space of tensors.

nk + k − 1 ≤ N

wheren = (

the variety of simple tensors;

N = (Πai )− 1 = dimension of the total space of tensors.

nk + k − 1 ≤ N

wheren = (

nk + k − 1 ≤ N

wheren = (

nk + k − 1 ≤ N

wheren = (

nk + k − 1 ≤ N

wheren = (

k-identifiability ⇔ ak is generically finite

impossible if dim(Aσk) > N.

nk + k − 1 ≤ N

wheren = (

The identifiabilty of a generic tensor of rank k and type a1 × · · · × asmakes sense only if

nk + k − 1 ≤ N

wheren = (

∑ai )− s = dimension of the Segre variety,

i.e.dimension of the variety of simple tensors;N = (Πai )− 1 = dimension of the total space of tensors.

Thus there exists a maximal value of the rank for which generick-identifiability is studied,

namely

kmax =(Πai )

ai )− s + 1

Indeed, the method of contact loci can only handle the case k < kmax .

nk + k − 1 ≤ N

wheren = (

Thus there exists a maximal value of the rank for which generick-identifiability is studied, namely

kmax =(Πai )

ai )− s + 1

nk + k − 1 ≤ N

wheren = (

Thus there exists a maximal value of the rank for which generick-identifiability is studied, namely

kmax =(Πai )

ai )− s + 1

General results

Case of tensors of type 2× · · · × 2 (s times).

Here kmax = 2s

s+1 .Theorems

(Catalisano - Geramita - Gimigliano (2002)) Non-defective, for s > 4, any rank k.

(Elmore - Hall - Neeman (2005)) k-identifiable, for s � k.

(Allman - Mathias - Rhodes (2009)) k-identifiable, for s > 5 and

k ≤√

2s−1 '√kmax/2.

(Bocci — (2011)) k-identifiable, for s > 5 and k ≤ 2s−1/s ' kmax/2.

(Bocci — Ottaviani (2013)) k-identifiable, for s > 5 and k ≤ (1023/1024)kmax .

(— Ottaviani - Vannieuwenhoven (2014)) k-identifiable, for s > 5 and

k ≤ (18383/18384)kmax .

General results

Case of tensors of type 2× · · · × 2 (s times). Here kmax = 2s

Theorems

k ≤√

2s−1 '√kmax/2.

k ≤ (18383/18384)kmax .

General results

s+1 .Theorems

k ≤√

2s−1 '√kmax/2.

k ≤ (18383/18384)kmax .

General results

s+1 .Theorems

(Elmore - Hall - Neeman (2005)) k-identifiable, for s � k .

k ≤√

2s−1 '√kmax/2.

k ≤ (18383/18384)kmax .

General results

s+1 .Theorems

k ≤√

2s−1

'√kmax/2.

k ≤ (18383/18384)kmax .

General results

s+1 .Theorems

k ≤√

2s−1 '√kmax/2.

k ≤ (18383/18384)kmax .

General results

s+1 .Theorems

k ≤√

2s−1 '√kmax/2.

(Bocci — (2011)) k-identifiable, for s > 5 and k ≤ 2s−1/s

' kmax/2.

k ≤ (18383/18384)kmax .

General results

s+1 .Theorems

k ≤√

2s−1 '√kmax/2.

k ≤ (18383/18384)kmax .

General results

s+1 .Theorems

k ≤√

2s−1 '√kmax/2.

k ≤ (18383/18384)kmax .

General results

s+1 .Theorems

k ≤√

2s−1 '√kmax/2.

k ≤ (18383/18384)kmax .

Tensors of type 2× · · · × 2 (s times)

Method:

Use Catalisano - Geramita - Gimigliano.

Translate to the non-existence of contact loci, to obtain results foridentifiability.

Method:

Method

Consider the Segre product (P1)s asa family of products (P1)s−1,parameterized by P1.

Push the k points to two copies of(P1)s−1.

Use induction to detect thenon-existence of contact loci.

Of course one should know how tohandle tangent spaces.

Method

Initial step

Why (1023/1024)?

It depends on the choice of the initial step.If one uses (P1)10 as the initial step, then the coefficient that one gets byinduction is (210 − 1)/210.

Yet one needs first to handle directly smaller cases.These was done via a computer algebra package (Macaulay2, symboliccomputations).

With a different approach one can improve the initial step of theinduction.

k ≤ (18383/18384)kmax .

Initial step

Why (1023/1024)?

k ≤ (18383/18384)kmax .

Initial step

Why (1023/1024)?

It depends on the choice of the initial step.

If one uses (P1)10 as the initial step, then the coefficient that one gets byinduction is (210 − 1)/210.

k ≤ (18383/18384)kmax .

Initial step

Why (1023/1024)?

k ≤ (18383/18384)kmax .

Initial step

Why (1023/1024)?

Yet one needs first to handle directly smaller cases.

These was done via a computer algebra package (Macaulay2, symboliccomputations).

k ≤ (18383/18384)kmax .

Initial step

Why (1023/1024)?

Yet one needs first to handle directly smaller cases.These was done via a computer algebra package

(Macaulay2, symboliccomputations).

k ≤ (18383/18384)kmax .

Initial step

Why (1023/1024)?

k ≤ (18383/18384)kmax .

Initial step

Why (1023/1024)?

k ≤ (18383/18384)kmax .

Conjecture

Theorem (— Ottaviani - Vannieuwenhoven (2014))

General tensors of type 2× · · · × 2 (s times) are k-identifiable, for s > 5and k ≤ (18383/18384)kmax .

Together they suggest:

Conjecture

General tensors of type 2× · · · × 2 (s times) are k-identifiable, for s > 5and any k < kmax .

Conjecture

Theorem (— Ottaviani - Vannieuwenhoven (2014))

General tensors of type 2× · · · × 2 (s times) are k-identifiable, for s > 5and k ≤ (18383/18384)kmax .

Together they suggest:

Conjecture

General tensors of type 2× · · · × 2 (s times) are k-identifiable, for s > 5and any k < kmax .

Tensors of type a × b × c

Some known results:

Kruskal’s Theorem (1977)

The general tensor of type a× b × c is k-identifiable for

k ≤ 1

2(min(a, k) + min(b, k) + min(c , k)− 2).

De Lathauwer’s Theorem for unbalanced tensors (2006)

The general tensor of type a× b × c is k-identifiable for k ≤ c andk(k − 1) ≤ a(a− 1)b(b − 1)/2.

Domanov - De Lathauwer (2014)

The general tensor of type a×b× c is k-identifiable for a ≤ b ≤ c ≤ k and

k ≤a + b + 2c − 2−

√(b − a)2 + 4c

Some known results:

k ≤ 1

k ≤a + b + 2c − 2−

√(b − a)2 + 4c

Some known results:

k ≤ 1

k ≤a + b + 2c − 2−

√(b − a)2 + 4c

Some known results:

k ≤ 1

k ≤a + b + 2c − 2−

√(b − a)2 + 4c

Some known results:

k ≤ 1

k ≤a + b + 2c − 2−

√(b − a)2 + 4c

The geometric methods apply for the study of the identifiability of generaltensors.

Strassen (1983)

The general tensor of type a× b × c , with a ≤ b ≤ c , is k-identifiable forc odd and

k ≤ abc

a + b + c − 2− c = kmax − c .

Bocci — Ottaviani (2013)

The previous statement holds for any c ≥ a, b.

Strassen (1983)

k ≤ abc

a + b + c − 2− c

= kmax − c .

Strassen (1983)

k ≤ abc

a + b + c − 2− c = kmax − c .

Strassen (1983)

k ≤ abc

a + b + c − 2− c = kmax − c .

Unbalanced tensors of type a × b × c

For tensors of type a× b × c , with a ≤ b ≤ c, define the unbalancedcase as the case

c > ab − a− b.

Unbalanced case: Bocci — Ottaviani (2013)

In the unbalanced case, a general tensor of rank k is identifiable if andonly if

k ≤ ab − a− b − 1.

In the balanced case, all products have been tested up to abc ≤ 50, 000(or maybe more, today ...).

c > ab − a− b.

k ≤ ab − a− b − 1.

c > ab − a− b.

k ≤ ab − a− b − 1.

In the balanced case, all products have been tested up to abc ≤ 50, 000

(or maybe more, today ...).

c > ab − a− b.

k ≤ ab − a− b − 1.

Other examples

In the balanced case, all products have been tested up to abc ≤ 50, 000.

Example: — Mella - Ottaviani (2014)

General tensors of rank 8 and type 3× 6× 6 are not identifiable.They have at least 6 different decompositions.The contact loci are fourfolds: they correspond to P2 × P1 × P1 embeddedwith divisors of type (1, 1, 3).

There are strong evidences that indeed a general tensor as above hasexactly 6 decompositions.

Are tensors of type 3× 6× 6 the biggest (balanced) example of genericnon-identifiability for tensors of type a× b × c?

Other examples

General tensors of rank 8 and type 3× 6× 6 are not identifiable.

They have at least 6 different decompositions.The contact loci are fourfolds: they correspond to P2 × P1 × P1 embeddedwith divisors of type (1, 1, 3).

Other examples

General tensors of rank 8 and type 3× 6× 6 are not identifiable.They have at least 6 different decompositions.

The contact loci are fourfolds: they correspond to P2 × P1 × P1 embeddedwith divisors of type (1, 1, 3).

Other examples

Tensors of type a1 × · · · × as

Most results as above can be extended to the case of higher-dimensionaltensors.

Cubic case: Bocci — Ottaviani (2013)

The general tensor of type a× · · · × a, s times, is k-identifiable for

k ≤ as − (3a− 2)as−2

as − s + 1' (a− 1)(a− 2)

a2kmax .

For a1 ≤ · · · ≤ as and as > Πs−1ai −∑s−1(ai − 1), a general tensor of rank k

is identifiable if and only if

k ≤ Πs−1ai − 1−s−1∑

(ai − 1).

k ≤ as − (3a− 2)as−2

as − s + 1' (a− 1)(a− 2)

a2kmax .

k ≤ Πs−1ai − 1−s−1∑

(ai − 1).

k ≤ as − (3a− 2)as−2

as − s + 1

' (a− 1)(a− 2)

a2kmax .

k ≤ Πs−1ai − 1−s−1∑

(ai − 1).

k ≤ as − (3a− 2)as−2

as − s + 1' (a− 1)(a− 2)

a2kmax .

k ≤ Πs−1ai − 1−s−1∑

(ai − 1).

k ≤ as − (3a− 2)as−2

as − s + 1' (a− 1)(a− 2)

a2kmax .

k ≤ Πs−1ai − 1−s−1∑

(ai − 1).

Some computational aspect

— Ottaviani - Vannieuwenhoven (2014)

General tensors of type 2× · · · × 2 (s times) are k-identifiable, fors > 5 and k ≤ (18383/18384)kmax .

In the balanced case, all products have been tested up toabc ≤ 50, 000.

Idea: control the tangent space to the contact locus.

Inside Segre varieties, tangent spacesare defined by coordinate linear spaces.

Stacked Hessians:

H11 . . . H1s

. . . . . . . . .Hs1 . . . Hss

(Hij)hk =∂2q

∂uih∂ujk

q = any linear equation for a k-tangent space,uih’s = local coordinates of the i-th factor of the Segre product.

Stacked Hessians:

H11 . . . H1s

. . . . . . . . .Hs1 . . . Hss

(Hij)hk =∂2q

∂uih∂ujk

Stacked Hessians:

H11 . . . H1s

. . . . . . . . .Hs1 . . . Hss

(Hij)hk =∂2q

∂uih∂ujk

Stacked Hessians:

H11 . . . H1s

. . . . . . . . .Hs1 . . . Hss

(Hij)hk =∂2q

∂uih∂ujk

q = any linear equation for a k-tangent space,

uih’s = local coordinates of the i-th factor of the Segre product.

Stacked Hessians:

H11 . . . H1s

. . . . . . . . .Hs1 . . . Hss

(Hij)hk =∂2q

∂uih∂ujk

Stacked Hessians

H11 . . . H1s

. . . . . . . . .Hs1 . . . Hss

(Hij)hk =∂2q

∂aih∂ajk

The existence of the contact locus can be locally tested by means of the stackedHessian.

If the rank of the stacked Hessians is maximal at one point of the decompositionof a general tensor of rank k, then generic k-identifiability holds.

In this setting, computations are performed by linear algebra packages instead of

symbolic algebra packages (and linear algebra packages are much faster).

Stacked Hessians

H11 . . . H1s

. . . . . . . . .Hs1 . . . Hss

(Hij)hk =∂2q

∂aih∂ajk

If the rank of the stacked Hessians is maximal

at one point of the decompositionof a general tensor of rank k, then generic k-identifiability holds.

Stacked Hessians

H11 . . . H1s

. . . . . . . . .Hs1 . . . Hss

(Hij)hk =∂2q

∂aih∂ajk

If the rank of the stacked Hessians is maximal at one point of the decompositionof a general tensor of rank k,

then generic k-identifiability holds.

Stacked Hessians

H11 . . . H1s

. . . . . . . . .Hs1 . . . Hss

(Hij)hk =∂2q

∂aih∂ajk

Stacked Hessians

H11 . . . H1s

. . . . . . . . .Hs1 . . . Hss

(Hij)hk =∂2q

∂aih∂ajk

symbolic algebra packages

(and linear algebra packages are much faster).

Stacked Hessians

H11 . . . H1s

. . . . . . . . .Hs1 . . . Hss

(Hij)hk =∂2q

∂aih∂ajk

Specific tensors

The methods of Kruskal or De Lathauwer are able to detect theidentifiability not only of generic tensors

but also of specific given tensors.

Also the method of stacked Hessians can be used to detect theidentifiability of some specific tensors.

Zariski Main Theorem

Assume that a tensor T is a smooth point of the secant variety.If the general fiber ak : Aσk → PN isgiven by permutations

i.e. if the generic tensor is identifiable

and the fiber over a specific point Tis not a singleton (mod permut.)

i.e. some specific tensor T is notidentifiable

then the fiber over T is positivedimensional

i.e. a contact locus exists at somepoint of the decomposition.

Specific tensors

The methods of Kruskal or De Lathauwer are able to detect theidentifiability not only of generic tensorsbut also of specific given tensors.

Specific tensors

Assume that a tensor T is a smooth point of the secant variety.

If the general fiber ak : Aσk → PN isgiven by permutations

Specific tensors

i.e. a contact locus exists

at somepoint of the decomposition.

Specific tensors

One example of specific identifiability

It is not easy to detect the smoothness of a tensor T in a k-secantvariety.

But when it is possible, the method of Stacked Hessian provides aproof of the identifiability of T .

Example: An identifiable tensor of rank 7 and type 5× 5× 5.

The smoothness can be tested because one knows many local equations ofσ7(P4 × P4 × P4).The identifiability can now be proved via the stacked Hessian.

This case lies beyond Kruskal’s range.

It is not easy to detect the smoothness of a tensor T in a k-secantvariety. But when it is possible, the method of Stacked Hessian provides aproof of the identifiability of T .

The smoothness can be tested because one knows many local equations ofσ7(P4 × P4 × P4).

The identifiability can now be proved via the stacked Hessian.

Symmetric tensors

The method of stacked Hessian can be used also for the case of symmetrictensors

(and symmetric rank, symmetric identifiability).

Example

A general symmetric tensor of type 6× 6× 6 is not k-identifiable for k = 9(= bkmaxc).The contact locus is an elliptic normal curve in P5.

(Not) well known (Veneroni 1905 - Room).

Symmetric tensors

The method of stacked Hessian can be used also for the case of symmetrictensors (and symmetric rank, symmetric identifiability).

Example

Symmetric tensors

Example

A general symmetric tensor of type 6× 6× 6 is not k-identifiable for k = 9

(= bkmaxc).The contact locus is an elliptic normal curve in P5.

Symmetric tensors

Example

Symmetric tensors

Example

Cubics

General factSymmetric tensors of type n × n are almost never identifiable.

Theorem (Ballico, 2006)

General symmetric tensors of type n × · · · × n, s times s > 3, are identifiable, forany k < kmax , except for few cases, classified.

It remains to understand the case of symmetric tensors n × n × n (cubics in Pn).

Conjecture

For n > 6, cubics of any rank k are generically identifiable.

An inductive method of Brambilla - Ottaviani provides a proof of theconjecture, except for the case n = 3m + 2, k = bkmaxc.

W O R K I N P R O G R E S S

Cubics

General symmetric tensors of type n × · · · × n, s times s > 3, are identifiable, forany k < kmax ,

except for few cases, classified.

Conjecture

Cubics

Conjecture

Cubics

It remains to understand the case of symmetric tensors n × n × n

(cubics in Pn).

Conjecture

Cubics

Conjecture

Cubics

Conjecture

Cubics

Conjecture

Cubics

Conjecture

Other perspectives

* Systematic study of the singularieties of secant varieties

ofhomogeneous varieties.

* Method of the Hilbert function to detect the identifiability of ageneral or specific symmetric tensor.

* Identifiability for linear systems of tensors.

* Other definitions of simple tensors.

... a world of theorems ...

Thank you for your attention.

Other perspectives

* Systematic study of the singularieties of secant varieties ofhomogeneous varieties.

Other perspectives

The Geometry of the Identi ability of Tensors · 2020-01-03 · Luca Chiantini (Universita’ di...

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