From moments to sparse representations, a geometric, algebraic and algorithmic viewpoint. Tensors (10-14 September 2018, Polytechnico di Torino) - From moments to sparse representations, a geometric, algebraic and algorithmic viewpoint. Part 2.
From moments to sparse representations, a geometric, algebraic and algorithmi...BernardMourrain
Tensors (10-14 September 2018, Polytechnico di Torino) - From moments to sparse representations, a geometric, algebraic and algorithmic viewpoint. Tensors (10-14 September 2018, Polytechnico di Torino) - From moments to sparse representations, a geometric, algebraic and algorithmic viewpoint. Part 1.
From moments to sparse representations, a geometric, algebraic and algorithmi...BernardMourrain
Tensors (10-14 September 2018, Polytechnico di Torino) - From moments to sparse representations, a geometric, algebraic and algorithmic viewpoint. Tensors (10-14 September 2018, Polytechnico di Torino) - From moments to sparse representations, a geometric, algebraic and algorithmic viewpoint. Part 1.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Mathematics and History of Complex VariablesSolo Hermelin
Mathematics of complex variables, plus history.
This presentation is at a Undergraduate in Science (Math, Physics, Engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com, thanks! For more presentations, please visit my website at http://www.solohermelin.com
x2 y2
Standard Equation of hyperbola is a 2 – b2 = 1
(i) Definition hyperbola : A Hyperbola is the locus of a point in a plane which moves in the plane in such a way that the ratio of its distance from a fixed point (called focus) in the same plane to its distance from a fixed line (called directrix) is always constant which is always greater than unity.
The hyperbola whose transverse and conjugate axes are respectively the conjugate and transverse axes of a given hyperbola is called conjugate hyperbola.
Note :
(i) If e1 and e2 are the eccentricities of the
(ii) Vertices : The point A and A where the curve meets the line joining the foci S and S
hyperbola and its conjugate then
1 +
e 2 e
1 = 1
2
are called vertices of hyperbola.
(iii) Transverse and Conjugate axes : The straight line joining the vertices A and A is called transverse axes of the hyperbola. Straight line perpendicular to the transverse axes and passes through its centre called conjugate axes.
(iv) Latus Rectum : The chord of the hyperbola which passes through the focus and is perpendicular to its transverse axes is called
2b2
latus rectum. Length of latus rectum = a .
(ii) The focus of hyperbola and its1 conju2gate are concyclic.
Standard Equation and Difinitions
Ex.1 Find the equation of the hyperbola whose directrix is 2x + y = 1, focus (1,2) and
eccentricity 3 .
Sol. Let P (x,y) be any point on the hyperbola. Draw PM perpendicular from P on the directrix.
Then by definition SP = e PM
(v) Eccentricity : For the hyperbola
x2 y2
a 2 – b2
= 1,
(SP)2 = e2(PM)2
2x y 12
b2 = a2 (e2 – 1)
(x–1)2 + (y–2)2 = 3
Conjugate axes 2
5(x2 + y2 – 2x – 4y + 5} =
e = =
1
Transverse
axes
3(4x2 + y2 + 1+ 4xy – 2y – 4x)
7x2 – 2y2 + 12xy – 2x + 14y – 22 = 0
(vi) Focal distance : The distance of any point on the hyperbola from the focus is called the focal distance of the point.
Note : The difference of the focal distance of a point on the hyperbola is constant and is equal to the length
of the transverse axes. |SP – SP| = 2a (const.)
which is the required hyperbola.
Ex.2 Find the lengths of transverse axis and conjugate axis, eccentricity and the co- ordinates of foci and vertices; lengths of the latus rectum, equations of the directrices of the hyperbola 16x2 – 9y2 = –144
Sol. The equation 16x2 – 9y2 = – 144 can be
Sol. y= m1(x –a),y= m2(x + a) where m1m2 = k, given
x 2
written as 9
x2
y 2
– 16 = – 1. This is of the form
y2
In order to find the locus of their point of intersection we have to eliminate the unknown
m1 and m2. Multiplying, we get
y2 = m1m2 (x2 – a2) or y2 = k(x2–a2)
a 2 – b2 = – 1
a2 = 9, b2 = 16 a = 3, b = 4
or x – y
1 k
= a2
which represents a hyperbola.
Length of transverse axis :
The length of transverse axis = 2b = 8
Length of conjugate axis :
The length of conjugate axis = 2a = 6
5
Ex.5 T
II PUC (MATHEMATICS) ANNUAL MODEL QUESTION PAPER FOR ALL SCIENCE STUDENTS WHO...Bagalkot
My dear Students,
Wishing you all happy SHIVRATRI. & ALL THE BEST IN YOUR ANNUAL EXAMS-2014
Here I have uploaded II- P.U.C MATHEMATICS MODEL QUESTION PAPER FOR the year 2014 Which i have designed according to New syllabus of CBSE. I hope this model paper will be helpful to all the students who are writing annual exams on 18-March-2014.
wish you all the best
Regards,
A. NAGARAJ
Director-Faculty
Shree Susheela Tutorials
BAGALKOT-587101
mob: 9845222682
I am Frank P. I am a Physical Chemistry Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Physical Chemistry, from Malacca, Malaysia. I have been helping students with their homework for the past 6 years. I solve assignments related to Physical Chemistry.
Visit eduassignmenthelp.com or email info@eduassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Physical Chemistry Assignments.
All of material inside is un-licence, kindly use it for educational only but please do not to commercialize it.
Based on 'ilman nafi'an, hopefully this file beneficially for you.
Thank you.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Mathematics and History of Complex VariablesSolo Hermelin
Mathematics of complex variables, plus history.
This presentation is at a Undergraduate in Science (Math, Physics, Engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com, thanks! For more presentations, please visit my website at http://www.solohermelin.com
x2 y2
Standard Equation of hyperbola is a 2 – b2 = 1
(i) Definition hyperbola : A Hyperbola is the locus of a point in a plane which moves in the plane in such a way that the ratio of its distance from a fixed point (called focus) in the same plane to its distance from a fixed line (called directrix) is always constant which is always greater than unity.
The hyperbola whose transverse and conjugate axes are respectively the conjugate and transverse axes of a given hyperbola is called conjugate hyperbola.
Note :
(i) If e1 and e2 are the eccentricities of the
(ii) Vertices : The point A and A where the curve meets the line joining the foci S and S
hyperbola and its conjugate then
1 +
e 2 e
1 = 1
2
are called vertices of hyperbola.
(iii) Transverse and Conjugate axes : The straight line joining the vertices A and A is called transverse axes of the hyperbola. Straight line perpendicular to the transverse axes and passes through its centre called conjugate axes.
(iv) Latus Rectum : The chord of the hyperbola which passes through the focus and is perpendicular to its transverse axes is called
2b2
latus rectum. Length of latus rectum = a .
(ii) The focus of hyperbola and its1 conju2gate are concyclic.
Standard Equation and Difinitions
Ex.1 Find the equation of the hyperbola whose directrix is 2x + y = 1, focus (1,2) and
eccentricity 3 .
Sol. Let P (x,y) be any point on the hyperbola. Draw PM perpendicular from P on the directrix.
Then by definition SP = e PM
(v) Eccentricity : For the hyperbola
x2 y2
a 2 – b2
= 1,
(SP)2 = e2(PM)2
2x y 12
b2 = a2 (e2 – 1)
(x–1)2 + (y–2)2 = 3
Conjugate axes 2
5(x2 + y2 – 2x – 4y + 5} =
e = =
1
Transverse
axes
3(4x2 + y2 + 1+ 4xy – 2y – 4x)
7x2 – 2y2 + 12xy – 2x + 14y – 22 = 0
(vi) Focal distance : The distance of any point on the hyperbola from the focus is called the focal distance of the point.
Note : The difference of the focal distance of a point on the hyperbola is constant and is equal to the length
of the transverse axes. |SP – SP| = 2a (const.)
which is the required hyperbola.
Ex.2 Find the lengths of transverse axis and conjugate axis, eccentricity and the co- ordinates of foci and vertices; lengths of the latus rectum, equations of the directrices of the hyperbola 16x2 – 9y2 = –144
Sol. The equation 16x2 – 9y2 = – 144 can be
Sol. y= m1(x –a),y= m2(x + a) where m1m2 = k, given
x 2
written as 9
x2
y 2
– 16 = – 1. This is of the form
y2
In order to find the locus of their point of intersection we have to eliminate the unknown
m1 and m2. Multiplying, we get
y2 = m1m2 (x2 – a2) or y2 = k(x2–a2)
a 2 – b2 = – 1
a2 = 9, b2 = 16 a = 3, b = 4
or x – y
1 k
= a2
which represents a hyperbola.
Length of transverse axis :
The length of transverse axis = 2b = 8
Length of conjugate axis :
The length of conjugate axis = 2a = 6
5
Ex.5 T
II PUC (MATHEMATICS) ANNUAL MODEL QUESTION PAPER FOR ALL SCIENCE STUDENTS WHO...Bagalkot
My dear Students,
Wishing you all happy SHIVRATRI. & ALL THE BEST IN YOUR ANNUAL EXAMS-2014
Here I have uploaded II- P.U.C MATHEMATICS MODEL QUESTION PAPER FOR the year 2014 Which i have designed according to New syllabus of CBSE. I hope this model paper will be helpful to all the students who are writing annual exams on 18-March-2014.
wish you all the best
Regards,
A. NAGARAJ
Director-Faculty
Shree Susheela Tutorials
BAGALKOT-587101
mob: 9845222682
I am Frank P. I am a Physical Chemistry Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Physical Chemistry, from Malacca, Malaysia. I have been helping students with their homework for the past 6 years. I solve assignments related to Physical Chemistry.
Visit eduassignmenthelp.com or email info@eduassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Physical Chemistry Assignments.
All of material inside is un-licence, kindly use it for educational only but please do not to commercialize it.
Based on 'ilman nafi'an, hopefully this file beneficially for you.
Thank you.
Professional air quality monitoring systems provide immediate, on-site data for analysis, compliance, and decision-making.
Monitor common gases, weather parameters, particulates.
ESR spectroscopy in liquid food and beverages.pptxPRIYANKA PATEL
With increasing population, people need to rely on packaged food stuffs. Packaging of food materials requires the preservation of food. There are various methods for the treatment of food to preserve them and irradiation treatment of food is one of them. It is the most common and the most harmless method for the food preservation as it does not alter the necessary micronutrients of food materials. Although irradiated food doesn’t cause any harm to the human health but still the quality assessment of food is required to provide consumers with necessary information about the food. ESR spectroscopy is the most sophisticated way to investigate the quality of the food and the free radicals induced during the processing of the food. ESR spin trapping technique is useful for the detection of highly unstable radicals in the food. The antioxidant capability of liquid food and beverages in mainly performed by spin trapping technique.
Comparing Evolved Extractive Text Summary Scores of Bidirectional Encoder Rep...University of Maribor
Slides from:
11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), Niš, 3-6 June 2024
Track: Artificial Intelligence
https://www.etran.rs/2024/en/home-english/
DERIVATION OF MODIFIED BERNOULLI EQUATION WITH VISCOUS EFFECTS AND TERMINAL V...Wasswaderrick3
In this book, we use conservation of energy techniques on a fluid element to derive the Modified Bernoulli equation of flow with viscous or friction effects. We derive the general equation of flow/ velocity and then from this we derive the Pouiselle flow equation, the transition flow equation and the turbulent flow equation. In the situations where there are no viscous effects , the equation reduces to the Bernoulli equation. From experimental results, we are able to include other terms in the Bernoulli equation. We also look at cases where pressure gradients exist. We use the Modified Bernoulli equation to derive equations of flow rate for pipes of different cross sectional areas connected together. We also extend our techniques of energy conservation to a sphere falling in a viscous medium under the effect of gravity. We demonstrate Stokes equation of terminal velocity and turbulent flow equation. We look at a way of calculating the time taken for a body to fall in a viscous medium. We also look at the general equation of terminal velocity.
Nutraceutical market, scope and growth: Herbal drug technologyLokesh Patil
As consumer awareness of health and wellness rises, the nutraceutical market—which includes goods like functional meals, drinks, and dietary supplements that provide health advantages beyond basic nutrition—is growing significantly. As healthcare expenses rise, the population ages, and people want natural and preventative health solutions more and more, this industry is increasing quickly. Further driving market expansion are product formulation innovations and the use of cutting-edge technology for customized nutrition. With its worldwide reach, the nutraceutical industry is expected to keep growing and provide significant chances for research and investment in a number of categories, including vitamins, minerals, probiotics, and herbal supplements.
Remote Sensing and Computational, Evolutionary, Supercomputing, and Intellige...University of Maribor
Slides from talk:
Aleš Zamuda: Remote Sensing and Computational, Evolutionary, Supercomputing, and Intelligent Systems.
11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), Niš, 3-6 June 2024
Inter-Society Networking Panel GRSS/MTT-S/CIS Panel Session: Promoting Connection and Cooperation
https://www.etran.rs/2024/en/home-english/
BREEDING METHODS FOR DISEASE RESISTANCE.pptxRASHMI M G
Plant breeding for disease resistance is a strategy to reduce crop losses caused by disease. Plants have an innate immune system that allows them to recognize pathogens and provide resistance. However, breeding for long-lasting resistance often involves combining multiple resistance genes
This presentation explores a brief idea about the structural and functional attributes of nucleotides, the structure and function of genetic materials along with the impact of UV rays and pH upon them.
The use of Nauplii and metanauplii artemia in aquaculture (brine shrimp).pptxMAGOTI ERNEST
Although Artemia has been known to man for centuries, its use as a food for the culture of larval organisms apparently began only in the 1930s, when several investigators found that it made an excellent food for newly hatched fish larvae (Litvinenko et al., 2023). As aquaculture developed in the 1960s and ‘70s, the use of Artemia also became more widespread, due both to its convenience and to its nutritional value for larval organisms (Arenas-Pardo et al., 2024). The fact that Artemia dormant cysts can be stored for long periods in cans, and then used as an off-the-shelf food requiring only 24 h of incubation makes them the most convenient, least labor-intensive, live food available for aquaculture (Sorgeloos & Roubach, 2021). The nutritional value of Artemia, especially for marine organisms, is not constant, but varies both geographically and temporally. During the last decade, however, both the causes of Artemia nutritional variability and methods to improve poorquality Artemia have been identified (Loufi et al., 2024).
Brine shrimp (Artemia spp.) are used in marine aquaculture worldwide. Annually, more than 2,000 metric tons of dry cysts are used for cultivation of fish, crustacean, and shellfish larva. Brine shrimp are important to aquaculture because newly hatched brine shrimp nauplii (larvae) provide a food source for many fish fry (Mozanzadeh et al., 2021). Culture and harvesting of brine shrimp eggs represents another aspect of the aquaculture industry. Nauplii and metanauplii of Artemia, commonly known as brine shrimp, play a crucial role in aquaculture due to their nutritional value and suitability as live feed for many aquatic species, particularly in larval stages (Sorgeloos & Roubach, 2021).
ANAMOLOUS SECONDARY GROWTH IN DICOT ROOTS.pptxRASHMI M G
Abnormal or anomalous secondary growth in plants. It defines secondary growth as an increase in plant girth due to vascular cambium or cork cambium. Anomalous secondary growth does not follow the normal pattern of a single vascular cambium producing xylem internally and phloem externally.
Seminar of U.V. Spectroscopy by SAMIR PANDASAMIR PANDA
Spectroscopy is a branch of science dealing the study of interaction of electromagnetic radiation with matter.
Ultraviolet-visible spectroscopy refers to absorption spectroscopy or reflect spectroscopy in the UV-VIS spectral region.
Ultraviolet-visible spectroscopy is an analytical method that can measure the amount of light received by the analyte.
From moments to sparse representations, a geometric, algebraic and algorithmic viewpoint. B. Mourrain. Part 2
1. From moments to sparse representations,
a geometric, algebraic and algorithmic viewpoint
Bernard Mourrain
Inria M´editerran´ee, Sophia Antipolis
Bernard.Mourrain@inria.fr
Part II
2. Decomposition algorithms
1 Decomposition algorithms
2 Low rank classification
3 The variety of missing moments
B. Mourrain From moments to sparse representations 2 / 39
3. Decomposition algorithms
Univariate series:
Kronecker (1881)
The Hankel operator
Hσ : CN,finite → CN
(pm) → ( m σm+npm)n∈N
is of finite rank r iff ∃ω1, . . . , ωr ∈ C[y] and ξ1, . . . , ξr ∈ C distincts s.t.
σ(y) =
n∈N
σn
yn
n!
=
r
i=1
ωi (y)eξi
(y)
with r
i=1(deg(ωi ) + 1) = r.
B. Mourrain From moments to sparse representations 3 / 39
4. Decomposition algorithms
Multivariate series:
Theorem (Generalized Kronecker Theorem)
For σ = (σ1, . . . , σm) ∈ (R∗)m, the Hankel operator
Hσ : R → (R∗
)m
p → (p σ1, . . . , p σm)
is of rank r iff
σj =
r
i=1
ωj,i (y) eξi
(y) ∈ PolExp, j = 1, . . . , m
with r = r
i=1 µ(ω1,i , . . . , ωm,i ). In this case, we have
VC(Iσ) = {ξ1, . . . , ξr }.
Iσ = Q1 ∩ · · · ∩ Qr with Q⊥
i = ω1,i , . . . , ωm,i eξi
(y).
If m = 1, Aσ is Gorenstein (A∗
σ = Aσ σ is a free Aσ-module of rank 1)
and (a, b) → σ|ab is non-degenerate in Aσ.
B. Mourrain From moments to sparse representations 4 / 39
5. Decomposition algorithms
The decomposition from the algebraic structure
Decomposition problem
Given a (truncated) sequence of moments σα, α ∈ A, find
ξi = (ξi,1, ξi,1, . . . , ξi,n) ∈ K
n
disctint, ωi ∈ K. s.t. σ = i ωi eξi
Hankel operator: For σ ∈ R∗,
Hσ : R → R∗
p → p σ
Quotient algebra: Aσ = R/Iσ where Iσ = ker Hσ.
0 → Iσ → K[x]
Hσ
−→ A∗
σ → 0
p → p σ
Isomorphism between Aσ and A∗
σ = I⊥
σ .
(Aσ Gorenstein, i.e. ∃τ = σ ∈ A∗
σ s.t. A∗
σ = Aσ τ is a free Aσ-module).
6. Find the points ξi as the roots of Iσ and the weights ωi from the
idempotents of Aσ.
B. Mourrain From moments to sparse representations 5 / 39
7. Decomposition algorithms
The structure of Aσ
For σ = r
i=1 ωi eξi
, with ωi ∈ C {0} and ξi ∈ Cn distinct.
rank Hσ = r and the multiplicity of the points ξ1, . . . , ξr in V(Iσ) is 1.
For B, B be of size r, HB ,B
σ invertible iff B and B are bases of
Aσ = K[x]/Iσ.
The matrix Mi of multiplication by xi in the basis B of Aσ is such that
HB ,xiB
σ = HB ,B
xi σ = HB ,B
σ Mi
The common eigenvectors of Mi are (up to a scalar) the Lagrange
interpolation polynomials uξi
at the points ξi, i = 1, . . . , r.
uξi
(ξj)=
1 ifi = j,
0 otherwise
uξi
2 ≡ uξi
, r
i=1 uξi
≡ 1.
The common eigenvectors of Mt
i are (up to a scalar) the vectors
[B(ξi)], i = 1, . . . , r.
B. Mourrain From moments to sparse representations 6 / 39
8. Decomposition algorithms
Decomposition algorithm
Input: The first coefficients (σα)α∈A of the series
σ =
α∈Nn
σα
yα
α!
=
r
i=1
ωi eξi
(y)
1 Compute bases B, B ⊂ xA s.t. that HB ,B invertible and
|B| = |B | = r = dim Aσ;
2 Deduce the tables of multiplications Mi := (HB ,B
σ )−1HB ,xi B
σ
3 Compute the eigenvectors v1, . . . , vr of i li Mi for a generic
l = l1x1 + · · · + lnxn;
4 Deduce the points ξi = (ξi,1, . . . , ξi,n) s.t. Mj vi − ξi,j vi = 0 and the
weights ωi = 1
vi (ξi ) σ|vi .
Output: The decomposition σ = r
i=1
1
vi (ξi ) σ|vi eξi
(y).
B. Mourrain From moments to sparse representations 7 / 39
17. Its decomposition yields the Mi and the ancestor probabibility (πj ).
B. Mourrain From moments to sparse representations 14 / 39
18. Decomposition algorithms
A general framework
F the functional space, in which the “signal” lives.
S1, . . . , Sn : F → F commuting linear operators: Si ◦ Sj = Sj ◦ Si .
∆ : h ∈ F → ∆[h] ∈ C a linear functional on F.
Generating series associated to h ∈ F:
σh(y) =
α∈Nn
∆[Sα
(h)]
yα
α!
=
α∈Nn
σα
yα
α!
.
Eigenfunctions:
Sj (E) = ξj E, j = 1, . . . , n ⇒ σE = ω eξ(y).
Generalized eigenfunctions:
Sj (Ek ) = ξj Ek +
k <k
mj,k Ek ⇒ σEk
= ωi (y)eξ(y).
19. If h → σh is injective ⇒ unique decomposition of f as a linear
combination of generalized eigenfunctions.
B. Mourrain From moments to sparse representations 15 / 39
20. Decomposition algorithms
Sum of polynomial-exponential functions
F = PolExp(x),
Sj : h(x) → h(x1, . . . , xj−1, xj + δj , xj+1, . . . , xn) shift of xj by δj ,
∆ : h(x) → ∆[h] = h(0) the evaluation at 0.
Generating series of h: σh(y) = α∈Nn h(α1δ1, . . . , αnδn) yα
α! .
Eigenfunctions: ef·x; generalized eigenfunctions: ω(x)ef·x;
h(x) = r
i=1 gi (x)efi x + r(x) with gi (x) ∈ C[x], fi ∈ Cn and r(δ α) = 0,
∀α ∈ Nn, iff
σh(y) =
r
i=1
ωi (y)eξi
(y)
with ξi = efi ∈ V(ker Hσh
) ⊂ Cn , ωi (x) = α gi,αωα for gi = α gi,αxα.
21. Decomposition from the moments σα = h(α1δ1, . . . , αnδn).
B. Mourrain From moments to sparse representations 16 / 39
22. Decomposition algorithms
Sparse interpolation of PolyLog functions
F = PolyLog(x) = { (β,γ)∈A hβ,γ logβ
(x)xγ, A finite},
Sj : h(x1, . . . , xn) → h(. . . , xj−1, λj xj , xj+1, . . .) for λj ∈ C − {1},
∆ : h(x1, . . . , xn) → ∆[h] = h(1, . . . , 1).
Generating series of h: σh(y) = α∈Nn h(λα1
1 , . . . , λαn
n ) yα
α! .
Eigenfunctions: xγ; generalized eigenfunctions: logβ
(x)xγ.
h = r
i=1 β∈Bi
ωi,β logβ
(x) xγi iff the generating series σh is of the form
σh(y) =
r
i=1
ωi (y)eξi
(y)
with ξi = (λ
γi,1
1 , . . . , λ
γi,n
n ) ∈ Cn and ωi (y) = β∈Bi
ωi,βyβ ∈ C[y].
23. Decomposition from the moments σα = h(λα1
1 , . . . , λαn
n ).
B. Mourrain From moments to sparse representations 17 / 39
24. Decomposition algorithms
Sparse reconstruction from Fourier coefficients
F = L2(Ω);
Si : h(x) ∈ L2(Ω) → e
2π
xi
Ti h(x) ∈ L2(Ω) is the multiplication by e
2π
xi
Ti ;
∆ : h(x) ∈ OC → h(x)dx ∈ C.
The moments of f are
σγ =
1
n
j=1 Tj
f (x)e
−2πi n
j=1
γj xj
Tj dx
Eigenfunctions: δξ; generalized eigenfunctions: δ
(α)
ξ .
For f ∈ L2(Ω) and σ = (σγ)γ∈Zn its Fourier coefficients,
Γσ : (ρβ)β∈Zn ∈ L2
(Zn
) →
β
σα+βρβ
α∈Zn
∈ L2
(Zn
).
Γσ is of finite rank r if and only if f = r
i=1 α∈Ai ⊂Nn ωi,αδ
(α)
ξi
with
ξi = (ξi,1, . . . , ξi,n) ∈ Ω, ωi,α ∈ C and r = r
i=1 µ( α∈Ai
ωi,αyα)
B. Mourrain From moments to sparse representations 18 / 39
25. Decomposition algorithms
Other applications
Decomposition of measures as sums of spikes from moments (images,
spectroscopy, radar, astronomy, . . . )
Decomposition of convolution operators of finite rank
Vanishing ideal of points: σ = r
i=1 eξi
(y)
Change of ordering for Grobner bases or change of bases for
zero-dimensional ideals: σα = u, N(xα) ,
. . .
B. Mourrain From moments to sparse representations 19 / 39
26. Low rank classification
1 Decomposition algorithms
2 Low rank classification
3 The variety of missing moments
B. Mourrain From moments to sparse representations 20 / 39
27. Low rank classification
Low rank decomposition of Hankel matrices
Rank 1 Hankel matrices: Hξ = [ξα+β]α∈A,β∈B for some ξ ∈ Kn or Pn.
Rank r Hankel matrices are not necessarily the sum of r rank one Hankel
matrices.
0 1 0
1 0 0
0 0 0
= λ1
1 ξ1 ξ2
1
ξ1 ξ2
1 ξ3
1
ξ2
1 ξ3
1 ξ4
1
+ λ2
1 ξ2 ξ2
2
ξ2 ξ2
2 ξ3
2
ξ2
2 ξ3
2 ξ4
2
but
0 1 0
1 0 0
0 0 0
= lim
→0
1
2
1 2
2 3
2 3 4
−
1
2
1 − 2
− 2
− 3
2
− 3 4
Symbol: y = lim →0
1
2 (e y − e− y ).
B. Mourrain From moments to sparse representations 21 / 39
28. Low rank classification
Structured Low rank Decomposition
Decomposition in sum of Hankel operators associated to symbols
ω(y)eξ(y) with ω(y) ∈ K[y], ξ ∈ Cn.
30. σ = r
i=1 ωi (y) eξi
(y) ⇒
HA,B
σ = WA;Γ(ξ)∆Γ
ωWB;Γ(ξ)t
HA,B
g σ = WA;Γ(ξ)∆Γ
g ωWB;Γ(ξ)t
where WA;Γ(ξ) Wronskian, ∆Γ
g ω block diagonal.
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31. Low rank classification
Symmetric tensors of low rank (joint work with A. Oneto)
For ψ ∈ Sd of degree d, with a decomposition ψ = r
i=1 ξi , x d and for
0 ≤ k ≤ d − k,
Hd−k,k
ψ∗ = Vd−k(Ξ) V t
k (Ξ)
where Ξ = (ξ1, . . . , ξr ) ∈ (Kn+1)r , Vk(Ξ) is the Vandermonde matrix of Ξ
at the monomials of deg. k.
Notation
ψ⊥
k = ker Hd−k,k
ψ∗
h(k) = dim Sk/ψ⊥
k = rankHd−k,k
ψ∗
I(Ξ) defining ideal of the points Ξ
Apolarity lemma
Ξ is apolar to ψ (i.e. appears in a decomposition of ψ) iff I(Ξ)k ⊂ ψ⊥
k for
any k ∈ N.
Proof. ψ∗ ∈ eξ1 , . . . , eξr ⊂ S∗
d .
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32. Low rank classification
The regularity of Ξ is ρ(Ξ) = min{k ∈ N | ∃u1, . . . , ur ∈ Sk s.t. ui (ξj ) = δi,j }.
Regularity lemma
Let ψ ∈ Sd and let Ξ be a minimal set of points apolar to ψ. Then,
I(Ξ)k = ψ⊥
k for 0 ≤ k ≤ d − ρ(Ξ).
Proof. ψ⊥
k = ker Hd−k,k
ψ∗ = Vd−k(Ξ) V t
k (Ξ), I(Ξ)k = ker V t
k (Ξ) and
Vd−k(Ξ) injective for d − k ≥ ρ(Ξ).
Theorem
Let ψ ∈ Sd and let Ξ be a minimal set of points apolar to ψ. If
d ≥ 2ρ(Ξ) + 1, then
I(Ξ) = (ψ⊥
≤ρ(Ξ)+1).
Moreover, Ξ is the unique minimal set of points apolar to ψ.
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33. Low rank classification
A set of essential variables of ψ is a minimal set of linear
forms 1, . . . , N ∈ S, such that ψ ∈ C[ 1, . . . , N].
Proposition
[Car06] the number of essential variables is hψ(1);
[CC017] any minimal decomposition of ψ involves only linear forms
in the essential variables.
The Waring locus of ψ is the locus of linear forms that can appear in a
minimal decomposition of ψ, i.e.,
Wψ := [ ] ∈ P(S1) | ∃ 2, . . . , r , r = rank(ψ), s.t. ψ ∈ d
, d
2 , . . . , d
r ;
The complement is forbidden locus denoted Fψ := Pn Wψ.
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34. Low rank classification
Tensor with 2 essential variables (Sylvester method)
Let ψ(x0, x1) ∈ Sd = K[x0, x1]d .
The Hilbert function of Aψ∗ is of the form:
with (ψ⊥) = (G1, G2) of degree 0 ≤ d1 ≤ d2 ≤ d with d1 + d2 = d + 2.
If G1 has simple roots, then ψ is of rank d1 = deg(G1) and the roots
of G1 are the unique minimal set apolar to ψ.
Otherwise, ψ is of rank d2 = deg(G2) and Wψ is dense in P1. For a
generic choice of A ∈ Sd2−d1 , the roots of AG1 + G2 are a minimal set
apolar to ψ.
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35. Low rank classification
Cases of rank 4
(a)
Collinear
(b) Coplanar,
with 3 collinear.
(c) General
coplanar.
(d) General
points.
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36. Low rank classification
For ψ ∈ Sd of rank 4.
ψ has two essential variables (hψ(1) = 2):
ψ⊥ = (L1, . . . , Ln−1, G1, G2), where deg(Gi ) = di and
d1 ≤ d2. In particular, it has to be d ≥ 4 and:
(i) if d = 4, 5, 6, then d2 = 4, and minimal apolar sets
of points are defined by ideals
I(Ξ) = (L1, . . . , Ln−1, HG1 + αG2), for a general
choice of H ∈ T6−d and α ∈ C;
(ii) if d ≥ 7, then d1 = 4 and the unique minimal
apolar set of points is given by
I(Ξ) = (L1, . . . , Ln−1, G1).
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37. Low rank classification
ψ has three essential variables (hψ(1) = 3) and a
minimal apolar set Ξ of type (b):
(i) if d = 3, then V(ψ⊥
2 ) = P + D, where P is a
reduced point and D is connected scheme of
length 2 whose linear span is a line LD;
Any minimal apolar set is of the type P ∪ Ξ , with
Ξ ⊂ LD;
(ii) if d = 4, then hψ(2) = 4, V(ψ⊥
2 ) = P ∪ L, where P
is a reduced point and L is a line not passing
through P;
Any minimal apolar set is of the type P ∪ Ξ ,
where Ξ ⊂ L.
(iii) if d ≥ 5, then hψ(2) = 4, V(ψ⊥
2 ) = P ∪ L, where P
is a reduced point and L is a line not passing
through P and (ψ⊥
3 ) defines the unique minimal
apolar set.
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38. Low rank classification
ψ has three essential variables (hψ(1) = 3) and a
minimal apolar set Ξ of type (c):
(i) if d = 3, then V(ψ⊥
2 ) = ∅ and Wψ is dense in the
plane of essential variables;
(ii) if d ≥ 4, there is a unique minimal apolar set of
points given by I(Ξ) = (ψ⊥
2 ).
ψ has four essential variables:
there is a unique minimal apolar set of points given by
I(Ξ) = (ψ⊥
2 ).
B. Mourrain From moments to sparse representations 30 / 39
40. Low rank classification
High rank and small forbidden locus
Definition: generic rank = rank of tensors on a dense open subset of
the set of tensors.
Theorem (Alexander, Hirschovitz, 1995)
The generic rank of a tensor in K[x0, . . . , xn]d is 1
n+1
n+d
d , except for
d = 2 and (n, d) ∈ {(2, 4), (3, 4), (4, 3), (4, 4)}.
Theorem (Oneto,·, 2018)
Let g be the generic rank of tensors of degree d in Pn. Let ψ ∈ Sd with
r = rank(ψ). If r > g, then Wψ is dense in Pn.
B. Mourrain From moments to sparse representations 32 / 39
41. The variety of missing moments
1 Decomposition algorithms
2 Low rank classification
3 The variety of missing moments
B. Mourrain From moments to sparse representations 33 / 39
42. The variety of missing moments
Flat extension of a truncated moment matrix
For (monomial) sets B ⊂ C, B ⊂ C , B = C B, B = C B .
HC,C
σ = σ | xα+β
α∈C,β∈C
=
HB,B
σ HB,B
σ
HB,B
σ HB,B
σ
,
when
rank HC,C
σ = rank HB,B
σ
For B ⊂ K[x], let B+ = B ∪ x1B · · · xnB, ∂B = B+ B.
Theorem
Assume HB,B
σ invertible with |B| = |B | = r and C ⊃ B+, C ⊃ B +
connected to 1 (m ∈ C ⇒ m = 1 or m = xj m with m ∈ C).
HC,C
σ is a flat extension of HB,B
σ
⇔ The operators Mj := H
B,xj B
σ (HB,B
σ )−1 commute.
⇔ ∃! ˜σ ∈ PolExp s.t. rankH˜σ = r and ˜σ|C·C = σ.
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45. The variety of missing moments
General decomposition algorithm [BCMT10], [BS18]
Perform a generic change of coordinates ψ (x) = ψ(T x).
For r = max rankHk,d−k
σ ,. . .
For bases B, B of size r, connected to 1 (e.g. B stable by
division/Borel fixed stable by division);
1 Find the (unknown) moments of HB +
,B+
Λ s.t.
· HB ,B
Λ invertible and
· the operators Mi = Hxi B ,B
Λ (HB ,B
Λ )−1
commute.
2 Deduce the decomposition of σ (Algorithm 1).
3 If the roots are simple and the decomposition is valid for the moments
of ψ, stop and output a decomposition of ψ;
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46. The variety of missing moments
Challenges, open questions
Numerical stability, correction of errors,
Efficient construction of basis, complexity,
Super-resolution, collision of points,
Super-extrapolation,
Best low rank approximation,
...
Thanks for your attention
B. Mourrain From moments to sparse representations 38 / 39
47. The variety of missing moments
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Symmetric tensor decomposition.
Linear Algebra and Applications, 433(11-12):1851–1872,
2010.
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Reducing the number of variables of a polynomial.
In Algebraic geometry and geometric modeling, pages
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Archiv der Mathematik, 93(1):87–98, 2009.
Bernard Mourrain.
Polynomial-Exponential Decomposition from Moments.
Foundations of Computational Mathematics, 2017.
http://dx.doi.org/10.1007/s10208-017-9372-x.
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On minimal decompositions of low rank symmetric
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preprint, https://hal.archives-ouvertes.fr/hal-01803571,
May 2018.
B. Mourrain From moments to sparse representations 39 / 39