Non-negative CP decomposition of hyperspectral data

Miguel A. Veganzones, Jeremy Cohen, RodrigoCabral-Farias and Pierre Comon

gipsa-lab, CNRS, UMR 5216, F-38420, Grenoble, Francemiguel-angel.veganzones@gipsa-lab.fr

ERC AdG-2013-320594 DECODA

Outline

1 Motivation

2 Non-negative CP decomposition

3 Experimental results with time series

4 What about hyperspectral data cubes ?

Miguel A. Veganzones Compression-based NN-CP 2 / 21

Outline

1 Motivation

2 Non-negative CP decomposition

3 Experimental results with time series

4 What about hyperspectral data cubes ?

Miguel A. Veganzones Compression-based NN-CP 3 / 21

Goals

1. Analysis of hyperspectral data by means of CanonicalPolyadic (CP) tensor decomposition.1.1 Time-series.1.2 Multi-angle acquisitions.1.3 Conventional images.

2. Physical interpretation of the one-rank factors in terms ofspectral unmixing.

3. Applications :3.1 Snow cover maps of the Alps : collaboration with LTHE

laboratory and MeteoFrance.3.2 Analysis of Martian surface : collaboration with IPAG

(Mars-ReCo project).

Miguel A. Veganzones Compression-based NN-CP 4 / 21

Goals

1. Analysis of hyperspectral data by means of CanonicalPolyadic (CP) tensor decomposition.1.1 Time-series.1.2 Multi-angle acquisitions.1.3 Conventional images.

2. Physical interpretation of the one-rank factors in terms ofspectral unmixing.

3. Applications :3.1 Snow cover maps of the Alps : collaboration with LTHE

laboratory and MeteoFrance.3.2 Analysis of Martian surface : collaboration with IPAG

(Mars-ReCo project).

Miguel A. Veganzones Compression-based NN-CP 4 / 21

Goals

1. Analysis of hyperspectral data by means of CanonicalPolyadic (CP) tensor decomposition.1.1 Time-series.1.2 Multi-angle acquisitions.1.3 Conventional images.

2. Physical interpretation of the one-rank factors in terms ofspectral unmixing.

3. Applications :3.1 Snow cover maps of the Alps : collaboration with LTHE

laboratory and MeteoFrance.3.2 Analysis of Martian surface : collaboration with IPAG

(Mars-ReCo project).

Miguel A. Veganzones Compression-based NN-CP 4 / 21

Hyperspectral tensors

> Hyperspectral matrix : XI×J .• I : number of pixels (spatial way).• J : number of bands (spectral way).

> Hyperspectral tensor : X I×J×K .• K : number of time acquisitions (temporal way : SNOW

project).• K : number of angles (angular way : Mars-ReCo project).• What about data cubes (rows × columns × bands) ?

Miguel A. Veganzones Compression-based NN-CP 5 / 21

Hyperspectral tensors

> Hyperspectral matrix : XI×J .• I : number of pixels (spatial way).• J : number of bands (spectral way).

> Hyperspectral tensor : X I×J×K .• K : number of time acquisitions (temporal way : SNOW

project).• K : number of angles (angular way : Mars-ReCo project).• What about data cubes (rows × columns × bands) ?

Miguel A. Veganzones Compression-based NN-CP 5 / 21

Hyperspectral tensors

> Hyperspectral matrix : XI×J .• I : number of pixels (spatial way).• J : number of bands (spectral way).

> Hyperspectral tensor : X I×J×K .• K : number of time acquisitions (temporal way : SNOW

project).• K : number of angles (angular way : Mars-ReCo project).• What about data cubes (rows × columns × bands) ?

Miguel A. Veganzones Compression-based NN-CP 5 / 21

Outline

1 Motivation

2 Non-negative CP decomposition

3 Experimental results with time series

4 What about hyperspectral data cubes ?

Miguel A. Veganzones Compression-based NN-CP 6 / 21

Non-negative CP decomposition

> Formulation :

Xijk ≈R∑

r=1

AirBjrCkrλr. (1)

• R ∈ N+ : tensor non-negative rank.• AI×R : spatial factors.• BJ×R : spectral factors.• CK×R : temporal/angular factors.• ΛR×R : scaling diagonal matrix.• Everything is non-negative !

> Compact representation : X ≈ (A,B,C)Λ.> Optimization problem :

minimize ‖X − (A,B,C)Λ‖2Fw.r.t. A,B,C,Λ

subject to A � 0,B � 0,C � 0

(2)

Miguel A. Veganzones Compression-based NN-CP 7 / 21

Non-negative CP decomposition

> Formulation :

Xijk ≈R∑

r=1

AirBjrCkrλr. (1)

• R ∈ N+ : tensor non-negative rank.• AI×R : spatial factors.• BJ×R : spectral factors.• CK×R : temporal/angular factors.• ΛR×R : scaling diagonal matrix.• Everything is non-negative !

> Compact representation : X ≈ (A,B,C)Λ.> Optimization problem :

minimize ‖X − (A,B,C)Λ‖2Fw.r.t. A,B,C,Λ

subject to A � 0,B � 0,C � 0

(2)

Miguel A. Veganzones Compression-based NN-CP 7 / 21

Non-negative CP decomposition

> Formulation :

Xijk ≈R∑

r=1

AirBjrCkrλr. (1)

• R ∈ N+ : tensor non-negative rank.• AI×R : spatial factors.• BJ×R : spectral factors.• CK×R : temporal/angular factors.• ΛR×R : scaling diagonal matrix.• Everything is non-negative !

> Compact representation : X ≈ (A,B,C)Λ.> Optimization problem :

minimize ‖X − (A,B,C)Λ‖2Fw.r.t. A,B,C,Λ

subject to A � 0,B � 0,C � 0

(2)

Miguel A. Veganzones Compression-based NN-CP 7 / 21

Properties

> Best lower non-negative rank approximates always exist→The problem is well-posed.

> There exist upper bounds to the tensor rank, Ro, that ensureuniqueness -> But only for exact decompositions !

> (Recently proved) If condition R ≤ Ro holds true→ Almostalways the best lower non-negative rank approximate isunique.

Miguel A. Veganzones Compression-based NN-CP 8 / 21

Properties

> Best lower non-negative rank approximates always exist→The problem is well-posed.

> There exist upper bounds to the tensor rank, Ro, that ensureuniqueness -> But only for exact decompositions !

> (Recently proved) If condition R ≤ Ro holds true→ Almostalways the best lower non-negative rank approximate isunique.

Miguel A. Veganzones Compression-based NN-CP 8 / 21

Properties

> Best lower non-negative rank approximates always exist→The problem is well-posed.

> There exist upper bounds to the tensor rank, Ro, that ensureuniqueness -> But only for exact decompositions !

> (Recently proved) If condition R ≤ Ro holds true→ Almostalways the best lower non-negative rank approximate isunique.

Miguel A. Veganzones Compression-based NN-CP 8 / 21

Properties

> Best lower non-negative rank approximates always exist→The problem is well-posed.

> There exist upper bounds to the tensor rank, Ro, that ensureuniqueness -> But only for exact decompositions !

> (Recently proved) If condition R ≤ Ro holds true→ Almostalways the best lower non-negative rank approximate isunique.

Miguel A. Veganzones Compression-based NN-CP 8 / 21

Algorithms

> The CP optimization problem is highly non-convex.> Yet many algorithms provide rather precise computation.> These algorithms can be divided into two main classes :

• All-at-once gradient-based descent, e.g. : all CP parametersare updated at the same time using a gradient scheme andnon-negativity constraints are implemented through barriersor soft penalizations.• Alternating minimization : the cost function is minimized in an

alternating way for each factor (A, B or C) while the othersare fixed.

> Drawback : These algorithms are computationally expensive→ Useless for large tensors.• Proposed solution : compression-based NN-CP

decomposition.

Miguel A. Veganzones Compression-based NN-CP 9 / 21

Algorithms

> The CP optimization problem is highly non-convex.> Yet many algorithms provide rather precise computation.> These algorithms can be divided into two main classes :

• All-at-once gradient-based descent, e.g. : all CP parametersare updated at the same time using a gradient scheme andnon-negativity constraints are implemented through barriersor soft penalizations.• Alternating minimization : the cost function is minimized in an

alternating way for each factor (A, B or C) while the othersare fixed.

> Drawback : These algorithms are computationally expensive→ Useless for large tensors.• Proposed solution : compression-based NN-CP

decomposition.

Miguel A. Veganzones Compression-based NN-CP 9 / 21

Algorithms

> The CP optimization problem is highly non-convex.> Yet many algorithms provide rather precise computation.> These algorithms can be divided into two main classes :

• All-at-once gradient-based descent, e.g. : all CP parametersare updated at the same time using a gradient scheme andnon-negativity constraints are implemented through barriersor soft penalizations.• Alternating minimization : the cost function is minimized in an

alternating way for each factor (A, B or C) while the othersare fixed.

> Drawback : These algorithms are computationally expensive→ Useless for large tensors.• Proposed solution : compression-based NN-CP

decomposition.

Miguel A. Veganzones Compression-based NN-CP 9 / 21

Algorithms

> The CP optimization problem is highly non-convex.> Yet many algorithms provide rather precise computation.> These algorithms can be divided into two main classes :

• All-at-once gradient-based descent, e.g. : all CP parametersare updated at the same time using a gradient scheme andnon-negativity constraints are implemented through barriersor soft penalizations.• Alternating minimization : the cost function is minimized in an

alternating way for each factor (A, B or C) while the othersare fixed.

> Drawback : These algorithms are computationally expensive→ Useless for large tensors.• Proposed solution : compression-based NN-CP

decomposition.

Miguel A. Veganzones Compression-based NN-CP 9 / 21

Algorithms

> The CP optimization problem is highly non-convex.> Yet many algorithms provide rather precise computation.> These algorithms can be divided into two main classes :

• All-at-once gradient-based descent, e.g. : all CP parametersare updated at the same time using a gradient scheme andnon-negativity constraints are implemented through barriersor soft penalizations.• Alternating minimization : the cost function is minimized in an

alternating way for each factor (A, B or C) while the othersare fixed.

> Drawback : These algorithms are computationally expensive→ Useless for large tensors.• Proposed solution : compression-based NN-CP

decomposition.

Miguel A. Veganzones Compression-based NN-CP 9 / 21

Compression-based NN-CP decomposition

General approach [1] :1. Compress the original tensor X I,J,K on a small tensor

GR1,R2,R3 .• Supposition : the compressed tensor G contains almost the

same information as the original tensor X .2. Compute the CP decomposition of G.

• Consider non-negativity constraints of X in the optimizationproblem.• Estimate the compressed factors AR1×R

c , BR2×Rc and CR3×R

c .

3. Uncompress the estimated one-rank factors Ac, Bc and Cc

to obtain the original factors AI×R, BJ×R and CK×R.

[1] Cohen, J., Cabral-Farias, R., Comon, P. ; "Fast Decomposition of Large Nonnegative Tensors," IEEE SignalProcessing Letters, 22(7), pp. 862-866, 2015.

Miguel A. Veganzones Compression-based NN-CP 10 / 21

Compression-based NN-CP decomposition

General approach [1] :1. Compress the original tensor X I,J,K on a small tensor

GR1,R2,R3 .• Supposition : the compressed tensor G contains almost the

same information as the original tensor X .2. Compute the CP decomposition of G.

• Consider non-negativity constraints of X in the optimizationproblem.• Estimate the compressed factors AR1×R

c , BR2×Rc and CR3×R

c .

3. Uncompress the estimated one-rank factors Ac, Bc and Cc

to obtain the original factors AI×R, BJ×R and CK×R.

[1] Cohen, J., Cabral-Farias, R., Comon, P. ; "Fast Decomposition of Large Nonnegative Tensors," IEEE SignalProcessing Letters, 22(7), pp. 862-866, 2015.

Miguel A. Veganzones Compression-based NN-CP 10 / 21

Compression-based NN-CP decomposition

General approach [1] :1. Compress the original tensor X I,J,K on a small tensor

GR1,R2,R3 .• Supposition : the compressed tensor G contains almost the

same information as the original tensor X .2. Compute the CP decomposition of G.

• Consider non-negativity constraints of X in the optimizationproblem.• Estimate the compressed factors AR1×R

c , BR2×Rc and CR3×R

c .

3. Uncompress the estimated one-rank factors Ac, Bc and Cc

to obtain the original factors AI×R, BJ×R and CK×R.

[1] Cohen, J., Cabral-Farias, R., Comon, P. ; "Fast Decomposition of Large Nonnegative Tensors," IEEE SignalProcessing Letters, 22(7), pp. 862-866, 2015.

Miguel A. Veganzones Compression-based NN-CP 10 / 21

Outline

1 Motivation

2 Non-negative CP decomposition

3 Experimental results with time series

4 What about hyperspectral data cubes ?

Miguel A. Veganzones Compression-based NN-CP 11 / 21

Dataset

> 44 daily MODIS acquisitions with low cloud coverage.> Pre-processed to increase spatial resolution (250m).> Dimensions : 4800× 7× 44.

[2] Veganzones, M.A. ; Cohen, J. ; Cabral-Farias, R. ; Chanussot, J. ; Comon, P. ; "Nonnegative tensor CP decompositionof hyperspectral data," IEEE Transactions on Geoscience and Remote Sensing, 54(5), pp. 2577-2588, 2016.

Miguel A. Veganzones Compression-based NN-CP 12 / 21

Experimental methodology (I)

> Competing algorithms :• Compression-based CP algorithms : CCG and ProCoALS.• Conventional CP algorithm : ANLS.• Conventional day-basis full-constrained spectral unmixing

(FCLSU).> Groundtruth : 8 spectra (on-field acquisitions).

1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

Spectral Band

Radiance

medium snow

glacier+snow

old coarse

ice

debris

rocks

rain forest

pasture

Miguel A. Veganzones Compression-based NN-CP 13 / 21

Experimental methodology (II)

> We run 50 Monte Carlo runs for each of the algorithms for aset of different rank values in the range R ∈ [5, 15].

> For the compression-based CP algorithms, the compressedtensor, X c, has dimensions 175× 7× 25.

> Quality measures :• Reconstruction error : average RMSE.• Angular error w.r.t. the groundtruth spectral signatures.• Linear Pearson correlation w.r.t. the spatial abundances

obtained by the FCLSU.

Miguel A. Veganzones Compression-based NN-CP 14 / 21

Experimental methodology (II)

> We run 50 Monte Carlo runs for each of the algorithms for aset of different rank values in the range R ∈ [5, 15].

> For the compression-based CP algorithms, the compressedtensor, X c, has dimensions 175× 7× 25.

> Quality measures :• Reconstruction error : average RMSE.• Angular error w.r.t. the groundtruth spectral signatures.• Linear Pearson correlation w.r.t. the spatial abundances

obtained by the FCLSU.

Miguel A. Veganzones Compression-based NN-CP 14 / 21

Experimental methodology (II)

> We run 50 Monte Carlo runs for each of the algorithms for aset of different rank values in the range R ∈ [5, 15].

> For the compression-based CP algorithms, the compressedtensor, X c, has dimensions 175× 7× 25.

> Quality measures :• Reconstruction error : average RMSE.• Angular error w.r.t. the groundtruth spectral signatures.• Linear Pearson correlation w.r.t. the spatial abundances

obtained by the FCLSU.

Miguel A. Veganzones Compression-based NN-CP 14 / 21

Reconstruction errors

5 6 7 8 9 10 11 12 13 14 15

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6x 10

−4

Rank

Average Normalized RMSE

(a) ANLS5 6 7 8 9 10 11 12 13 14 15

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6x 10

−4

Rank

Average Normalized RMSE

(b) CCG

5 6 7 8 9 10 11 12 13 14 15

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6x 10

−4

Rank

Average Normalized RMSE

(c) ProCoALS

Miguel A. Veganzones Compression-based NN-CP 15 / 21

Spectral factors interpretation

5 6 7 8 9 10 11 12 13 14 150

5

10

15

20

25

30

35

Rank

Average angular distance

medium snow

glacier+snow

old coarse

ice

debris

rocks

rain forest

pasture

(a) ANLS

5 6 7 8 9 10 11 12 13 14 150

5

10

15

20

25

30

35

Rank

Average angular distance

medium snow

glacier+snow

old coarse

ice

debris

rocks

rain forest

pasture

(b) CCG

5 6 7 8 9 10 11 12 13 14 150

5

10

15

20

25

30

35

Rank

Average angular distance

medium snow

glacier+snow

old coarse

ice

debris

rocks

rain forest

pasture

(c) ProCoALS

Miguel A. Veganzones Compression-based NN-CP 16 / 21

Spatial factors interpretation

5 6 7 8 9 10 11 12 13 14 150

0.2

0.4

0.6

0.8

1

Rank

Average maximum correlation

medium snow

glacier+snow

old coarse

ice

debris

rocks

rain forest

pasture

(a) ANLS

5 6 7 8 9 10 11 12 13 14 150

0.2

0.4

0.6

0.8

1

Rank

Average maximum correlation

medium snow

glacier+snow

old coarse

ice

debris

rocks

rain forest

pasture

(b) CCG

5 6 7 8 9 10 11 12 13 14 150

0.2

0.4

0.6

0.8

1

Rank

Average maximum correlation

medium snow

glacier+snow

old coarse

ice

debris

rocks

rain forest

pasture

(c) ProCoALS

Miguel A. Veganzones Compression-based NN-CP 17 / 21

Spatial factors interpretation

> Best run with rank R = 8.

Permanent snow/ice

Seasonal snow/ice (type 1) Seasonal snow/ice (type 2)

Seasonal vegetation (type 1) Seasonal vegetation (type 2)

Miguel A. Veganzones Compression-based NN-CP 18 / 21

Outline

1 Motivation

2 Non-negative CP decomposition

3 Experimental results with time series

4 What about hyperspectral data cubes ?

Miguel A. Veganzones Compression-based NN-CP 19 / 21

Are hyperspectral data cubes tensors ?

> Hyperspectral images are often provided, not asnon-negative matrices, X ∈ RI×J

+ , but as data cubes,X ∈ RIr×Ic×J

+ , where Ir and Ic denote the number of rowsand columns respectively, being I = IrIc.

> It is possible to think of the data cube as a tensorrepresentation of the image, allowing to apply tensoranalysis techniques.

> This is fine !→ Employed for de-noising, compression, ...> ... but, in general, it is not possible to suppose a low rank,

that is, a small R.> Proposed solution : patch-tensors [3].

[3] Veganzones, M.A. ; Cohen, J. ; Cabral-Farias, R. ; Usevich, K., Drumetz, L. ; Chanussot, J. ; Comon, P. ; "CanonicalPolyadic decomposition of hyperspectral patch-tensors," 2016 EUSIPCO Conference (submitted).

Miguel A. Veganzones Compression-based NN-CP 20 / 21

Are hyperspectral data cubes tensors ?

> Hyperspectral images are often provided, not asnon-negative matrices, X ∈ RI×J

+ , but as data cubes,X ∈ RIr×Ic×J

+ , where Ir and Ic denote the number of rowsand columns respectively, being I = IrIc.

> It is possible to think of the data cube as a tensorrepresentation of the image, allowing to apply tensoranalysis techniques.

> This is fine !→ Employed for de-noising, compression, ...> ... but, in general, it is not possible to suppose a low rank,

that is, a small R.> Proposed solution : patch-tensors [3].

[3] Veganzones, M.A. ; Cohen, J. ; Cabral-Farias, R. ; Usevich, K., Drumetz, L. ; Chanussot, J. ; Comon, P. ; "CanonicalPolyadic decomposition of hyperspectral patch-tensors," 2016 EUSIPCO Conference (submitted).

Miguel A. Veganzones Compression-based NN-CP 20 / 21

Are hyperspectral data cubes tensors ?

> Hyperspectral images are often provided, not asnon-negative matrices, X ∈ RI×J

+ , but as data cubes,X ∈ RIr×Ic×J

+ , where Ir and Ic denote the number of rowsand columns respectively, being I = IrIc.

> It is possible to think of the data cube as a tensorrepresentation of the image, allowing to apply tensoranalysis techniques.

> This is fine !→ Employed for de-noising, compression, ...> ... but, in general, it is not possible to suppose a low rank,

that is, a small R.> Proposed solution : patch-tensors [3].

[3] Veganzones, M.A. ; Cohen, J. ; Cabral-Farias, R. ; Usevich, K., Drumetz, L. ; Chanussot, J. ; Comon, P. ; "CanonicalPolyadic decomposition of hyperspectral patch-tensors," 2016 EUSIPCO Conference (submitted).

Miguel A. Veganzones Compression-based NN-CP 20 / 21

Are hyperspectral data cubes tensors ?

> Hyperspectral images are often provided, not asnon-negative matrices, X ∈ RI×J

+ , but as data cubes,X ∈ RIr×Ic×J

+ , where Ir and Ic denote the number of rowsand columns respectively, being I = IrIc.

> It is possible to think of the data cube as a tensorrepresentation of the image, allowing to apply tensoranalysis techniques.

> This is fine !→ Employed for de-noising, compression, ...> ... but, in general, it is not possible to suppose a low rank,

that is, a small R.> Proposed solution : patch-tensors [3].

[3] Veganzones, M.A. ; Cohen, J. ; Cabral-Farias, R. ; Usevich, K., Drumetz, L. ; Chanussot, J. ; Comon, P. ; "CanonicalPolyadic decomposition of hyperspectral patch-tensors," 2016 EUSIPCO Conference (submitted).

Miguel A. Veganzones Compression-based NN-CP 20 / 21

Are hyperspectral data cubes tensors ?

> Hyperspectral images are often provided, not asnon-negative matrices, X ∈ RI×J

+ , but as data cubes,X ∈ RIr×Ic×J

+ , where Ir and Ic denote the number of rowsand columns respectively, being I = IrIc.

> It is possible to think of the data cube as a tensorrepresentation of the image, allowing to apply tensoranalysis techniques.

> This is fine !→ Employed for de-noising, compression, ...> ... but, in general, it is not possible to suppose a low rank,

that is, a small R.> Proposed solution : patch-tensors [3].

[3] Veganzones, M.A. ; Cohen, J. ; Cabral-Farias, R. ; Usevich, K., Drumetz, L. ; Chanussot, J. ; Comon, P. ; "CanonicalPolyadic decomposition of hyperspectral patch-tensors," 2016 EUSIPCO Conference (submitted).

Miguel A. Veganzones Compression-based NN-CP 20 / 21

Non-negative CP decomposition of hyperspectral data

Miguel A. Veganzones, Jeremy Cohen, RodrigoCabral-Farias and Pierre Comon

gipsa-lab, CNRS, UMR 5216, F-38420, Grenoble, Francemiguel-angel.veganzones@gipsa-lab.fr

ERC AdG-2013-320594 DECODA