Sparse Proteomics Analysis (SPA)

Sparse Proteomics Analysis (SPA) Toward a Mathematical Theory for Feature Selection from Forward Models Martin Genzel Technische Universität Berlin Winter School on Compressed Sensing December 5, 2015

Outline 1 Biological Background 2 Sparse Proteomics Analysis (SPA) 3 Theoretical Foundation by High-dimensional Estimation Theory Martin Genzel Sparse Proteomics Analysis (SPA) WiCoS 2015 2 / 19

1 Biological Background 2 Sparse Proteomics Analysis (SPA) 3 Theoretical Foundation by High-dimensional Estimation Theory

What is Proteomics? The pathological mechanisms of many diseases, such as cancer, are manifested on the level of protein activities. Martin Genzel Sparse Proteomics Analysis (SPA) WiCoS 2015 3 / 19

What is Proteomics? The pathological mechanisms of many diseases, such as cancer, are manifested on the level of protein activities. To improve clinical treatment options and early diagnostics, we need to understand protein structures and their interactions! Proteins are long chains of amino acids, controlling many biological and chemical processes in the human body. http://www.topsan.org/proteins/jcsg/3qxb Martin Genzel Sparse Proteomics Analysis (SPA) WiCoS 2015 3 / 19

What is Proteomics? The pathological mechanisms of many diseases, such as cancer, are manifested on the level of protein activities. To improve clinical treatment options and early diagnostics, we need to understand protein structures and their interactions! Proteins are long chains of amino acids, controlling many biological and chemical processes in the human body. The entire set of proteins at a certain point of time is called a proteome. Proteomics is the large-scale study of the human proteome. http://www.topsan.org/proteins/jcsg/3qxb Martin Genzel Sparse Proteomics Analysis (SPA) WiCoS 2015 3 / 19

What is Mass Spectrometry? How to capture a proteome? Martin Genzel Sparse Proteomics Analysis (SPA) WiCoS 2015 4 / 19

What is Mass Spectrometry? How to capture a proteome? Mass spectrometry (MS) is a popular technique to detect the abundance of proteins in samples (blood, urine, etc.). Martin Genzel Sparse Proteomics Analysis (SPA) WiCoS 2015 4 / 19

What is Mass Spectrometry? How to capture a proteome? Mass spectrometry (MS) is a popular technique to detect the abundance of proteins in samples (blood, urine, etc.). Schematic Work-Flow Laser Mass spectrum Sample + + + - - + + Detector Intensity (cts) Mass (m/z) Martin Genzel Sparse Proteomics Analysis (SPA) WiCoS 2015 4 / 19

Real-World MS-Data Intensity (cts) Mass (m/z) MS-vector: x = (x 1,..., x d ) R d, d 10 4... 10 6 Index ˆ= Mass/Feature, Entry ˆ= Intensity/Amplitude Martin Genzel Sparse Proteomics Analysis (SPA) WiCoS 2015 5 / 19

Feature Selection from MS-Data Goal: Detect a small set of features (disease fingerprint) that allows for an appropriate distinction between the diseased and healthy group. Schematic Work-Flow Samples Blood from healthy individual Blood from diseased individual Mass (m/z) Martin Genzel Sparse Proteomics Analysis (SPA) WiCoS 2015 6 / 19

Feature Selection from MS-Data Goal: Detect a small set of features (disease fingerprint) that allows for an appropriate distinction between the diseased and healthy group. Schematic Work-Flow Samples Mass Spectra Blood from healthy individual MS Blood from diseased individual MS Intensity (cts) Intensity (cts) Mass (m/z) Mass (m/z) Martin Genzel Sparse Proteomics Analysis (SPA) WiCoS 2015 6 / 19

Feature Selection from MS-Data Goal: Detect a small set of features (disease fingerprint) that allows for an appropriate distinction between the diseased and healthy group. Schematic Work-Flow Samples Mass Spectra Feature Selection Blood from healthy individual MS Intensity (cts) Mass (m/z) Disease Fingerprint Blood from diseased individual Comparing MS Intensity (cts) Mass (m/z) Martin Genzel Sparse Proteomics Analysis (SPA) WiCoS 2015 6 / 19

Mathematical Problem Formulation Supervised Learning: We are given n samples (x 1, y 1 ),..., (x n, y n ). x k R d : y k { 1, +1}: Mass spectrum of the k-th patient Health status of the k-th patient (healthy = +1, diseased = 1) Goal: Learn a feature vector ω R d which is sparse, i.e., few non-zero entries, ( stability, avoid overfitting) and its entries correspond to peaks that are highly correlated with the disease. ( interpretability, biological relevance) Martin Genzel Sparse Proteomics Analysis (SPA) WiCoS 2015 7 / 19

How to learn a fingerprint ω?

1 Biological Background 2 Sparse Proteomics Analysis (SPA) 3 Theoretical Foundation by High-dimensional Estimation Theory

Sparse Proteomics Analysis (SPA) Sparse Proteomics Analysis is a generic framework to meet this challenge. Martin Genzel Sparse Proteomics Analysis (SPA) WiCoS 2015 8 / 19

Sparse Proteomics Analysis (SPA) Sparse Proteomics Analysis is a generic framework to meet this challenge. Input: Sample pairs (x 1, y 1 ),..., (x n, y n ) R d { 1, +1} Compute: 1 Preprocessing (Smoothing, Standardization) 2 Feature Selection (LASSO, l 1 -SVM, Robust 1-Bit CS) 3 Postprocessing (Sparsification) Output: Sparse feature vector ω R d Biomarker extraction, dimension reduction Intensity (cts) Mass (m/z) Biomarker Identification Blood Sample Martin Genzel Sparse Proteomics Analysis (SPA) WiCoS 2015 8 / 19

Sparse Proteomics Analysis (SPA) Sparse Proteomics Analysis is a generic framework to meet this challenge. Input: Sample pairs (x 1, y 1 ),..., (x n, y n ) R d { 1, +1} Compute: 1 Preprocessing (Smoothing, Standardization) 2 Feature Selection (LASSO, l 1 -SVM, Robust 1-Bit CS) 3 Postprocessing (Sparsification) Output: Sparse feature vector ω R d Rest of this talk Martin Genzel Sparse Proteomics Analysis (SPA) WiCoS 2015 8 / 19

Feature Selection (Geometric Intuition) Linear Separation Model: Find a feature vector ω R d such that y k = sign( x k, ω ) for many k {1,..., n}. Moreover, ω should be sparse and interpretable. Martin Genzel Sparse Proteomics Analysis (SPA) WiCoS 2015 9 / 19

Feature Selection via the LASSO The LASSO (Tibshirani 96) n (y k x k, ω ) 2 subject to ω 1 R min ω R d k=1 Multivariate approach, originally designed for linear regression models: y k x k, ω, k = 1,..., n. But also applicable to non-linear models Next part Later: R s to allow for s-sparse solutions (with unit norm). Martin Genzel Sparse Proteomics Analysis (SPA) WiCoS 2015 10 / 19

Some Numerical Results 5-fold cross-validation for real-world pancreas data (156 samples): 1 Learn feature vector ω by SPA, using 80% of the samples. 2 Classify the remaining 20% of the sample by an ordinary SVM, after projecting onto supp(ω). 3 Iterate this procedure 12-times for random partitions. Classification accuracy for different sparsity levels s = # supp(ω) Martin Genzel Sparse Proteomics Analysis (SPA) WiCoS 2015 11 / 19

But what about theoretical guarantees?

1 Biological Background 2 Sparse Proteomics Analysis (SPA) 3 Theoretical Foundation by High-dimensional Estimation Theory

Toward a Theoretical Foundation of SPA Linear Separation Model: Explains the observations/labels: y k = sign( x k, ω 0 ), k = 1,..., n Martin Genzel Sparse Proteomics Analysis (SPA) WiCoS 2015 12 / 19

Toward a Theoretical Foundation of SPA Linear Separation Model: Explains the observations/labels: y k = sign( x k, ω 0 ), k = 1,..., n Forward Model: Explains the random distribution of the data: x k = M m=1 s m,ka m + n k, k = 1,..., n a m : s m,k : n k : Deterministic feature atom, sampled Gaussian peak ( R d ) Random latent factor specifying the peak amplitude ( R) Random baseline noise ( R d ) s ",$ s ",$ % exp (% c ") - β " - Martin Genzel Sparse Proteomics Analysis (SPA) WiCoS 2015 12 / 19

Toward a Theoretical Foundation of SPA Linear Separation Model: Explains the observations/labels: y k = sign( x k, ω 0 ), k = 1,..., n Forward Model: Explains the random distribution of the data: x k = M m=1 s m,ka m + n k, k = 1,..., n Supposed that sufficiently many samples are given, can we learn the sparse fingerprint ω 0? Problem: The vector ω 0 is not unique because some features are perfectly correlated No hope for support recovery or approximation Martin Genzel Sparse Proteomics Analysis (SPA) WiCoS 2015 12 / 19

Toward a Theoretical Foundation of SPA Linear Separation Model: Explains the observations/labels: y k = sign( x k, ω 0 ), k = 1,..., n Forward Model: Explains the random distribution of the data: x k = M m=1 s m,ka m + n k, k = 1,..., n Supposed that sufficiently many samples are given, can we learn the sparse fingerprint ω 0? Problem: The vector ω 0 is not unique because some features are perfectly correlated No hope for support recovery or approximation Idea: Separate the fingerprint from its data representation! Martin Genzel Sparse Proteomics Analysis (SPA) WiCoS 2015 12 / 19

Combining the Models Assumptions: x k = M m=1 s m,ka m + n k, k = 1,..., n s k := (s 1,k,..., s M,k ) N (0, I M ) peak amplitudes n k N (0, σ 2 I d ) noise vector a 1,..., a M R d arbitrary (peak) atoms, D := a 1. R M d Put this into the classification model: y k = sign( x k, ω 0 ) = sign( M m=1 s m,ka m + n k, ω 0 ) = sign( D s k + n k, ω 0 ) a M Martin Genzel Sparse Proteomics Analysis (SPA) WiCoS 2015 13 / 19

Combining the Models Assumptions: x k = M m=1 s m,ka m + n k, k = 1,..., n s k := (s 1,k,..., s M,k ) N (0, I M ) peak amplitudes n k N (0, σ 2 I d ) noise vector a 1,..., a M R d arbitrary (peak) atoms, D := a 1. R M d Put this into the classification model: y k = sign( x k, ω 0 ) = sign( M m=1 s m,ka m + n k, ω 0 ) a M = sign( D s k + n k, ω 0 ) = sign( s k, Dω 0 }{{} =:z 0 + n k, ω 0 ) = sign( s k, z 0 + n k, ω 0 ) Martin Genzel Sparse Proteomics Analysis (SPA) WiCoS 2015 13 / 19

Signal Space vs. Coefficient Space x k = M m=1 s m,ka m + n k = D s k + n k Martin Genzel Sparse Proteomics Analysis (SPA) WiCoS 2015 14 / 19

Signal Space vs. Coefficient Space x k = M m=1 s m,ka m = D s k Let us first assume that n k = 0 (no baseline noise). Martin Genzel Sparse Proteomics Analysis (SPA) WiCoS 2015 14 / 19

Signal Space vs. Coefficient Space x k = M m=1 s m,ka m = D s k Let us first assume that n k = 0 (no baseline noise). Then where z 0 = Dω 0. y k = sign( x k, ω 0 ) = sign( s k, z 0 ), Martin Genzel Sparse Proteomics Analysis (SPA) WiCoS 2015 14 / 19

Signal Space vs. Coefficient Space x k = M m=1 s m,ka m = D s k Let us first assume that n k = 0 (no baseline noise). Then where z 0 = Dω 0. y k = sign( x k, ω 0 ) = sign( s k, z 0 ), z 0 has a (non-unique) representation in the dictionary D with sparse coefficients ω 0. z 0 lives in the signal space R M (independent of specific data type). ω 0 lives in the coefficient space R d (data dependent). Martin Genzel Sparse Proteomics Analysis (SPA) WiCoS 2015 14 / 19

Signal Space vs. Coefficient Space x k = M m=1 s m,ka m = D s k Let us first assume that n k = 0 (no baseline noise). Then where z 0 = Dω 0. y k = sign( x k, ω 0 ) = sign( s k, z 0 ), z 0 has a (non-unique) representation in the dictionary D with sparse coefficients ω 0. z 0 lives in the signal space R M (independent of specific data type). ω 0 lives in the coefficient space R d (data dependent). Try to show a recovery result for z 0! Martin Genzel Sparse Proteomics Analysis (SPA) WiCoS 2015 14 / 19

What Does This Mean for the LASSO? y k = sign( x k, ω 0 ) = sign( s k, z 0 ) with z 0 = Dω 0 SPA via the LASSO n (y k x k, ω ) 2 subject to ω 1 R min ω R d k=1 Martin Genzel Sparse Proteomics Analysis (SPA) WiCoS 2015 15 / 19

What Does This Mean for the LASSO? y k = sign( x k, ω 0 ) = sign( s k, z 0 ) with z 0 = Dω 0 SPA via the LASSO n (y k x k, ω ) 2 min ω R B d 1 k=1 Martin Genzel Sparse Proteomics Analysis (SPA) WiCoS 2015 15 / 19

What Does This Mean for the LASSO? y k = sign( x k, ω 0 ) = sign( s k, z 0 ) with z 0 = Dω 0 SPA via the LASSO n z:=dω min (y k x k, ω ) 2 = min ω R B1 d }{{} z R DB k=1 1 d = s k,z n (y k s k, z ) 2 k=1 Martin Genzel Sparse Proteomics Analysis (SPA) WiCoS 2015 15 / 19

What Does This Mean for the LASSO? y k = sign( x k, ω 0 ) = sign( s k, z 0 ) with z 0 = Dω 0 SPA via the LASSO n (y k x k, ω ) 2 min ω R B d 1 k=1 } {{ } Solvable in practice! z:=dω = min z R DB1 d k=1 n (y k s k, z ) 2 } {{ } Solvable in theory! Warning: The minimizers live in different spaces! Warning: We neither know D nor s k, but just their product. Martin Genzel Sparse Proteomics Analysis (SPA) WiCoS 2015 15 / 19

What Does This Mean for the LASSO? y k = sign( x k, ω 0 ) = sign( s k, z 0 ) with z 0 = Dω 0 SPA via the LASSO n (y k x k, ω ) 2 min ω R B d 1 k=1 } {{ } Solvable in practice! z:=dω = min z R DB1 d k=1 n (y k s k, z ) 2 } {{ } Solvable in theory! Warning: The minimizers live in different spaces! Warning: We neither know D nor s k, but just their product. Idea: Apply results for the K-LASSO with K = R DB d 1! Martin Genzel Sparse Proteomics Analysis (SPA) WiCoS 2015 15 / 19

A Simplified Version of Roman Vershynin s Result Theorem (Plan, Vershynin 15) Suppose that s k N (0, I M ), z 0 S M 1, and the observations follow y k = sign( s k, z 0 ), k = 1,..., n. 2 Put µ = π and assume that µz 0 K, where K is convex, and n w(k) 2. Then, with high probability, the solution ẑ of the K-LASSO satisfies w(k) ẑ µz 0 2 n. The (global) mean width for bounded K R M is given by w(k) = sup g, u, where g N (0, I M ). u K Martin Genzel Sparse Proteomics Analysis (SPA) WiCoS 2015 16 / 19

A Simplified Version of Roman Vershynin s Result Theorem (Plan, Vershynin 15) Suppose that s k N (0, I M ), z 0 S M 1, and the observations follow y k = sign( s k, z 0 ), k = 1,..., n. 2 Put µ = π and assume that µz 0 K, where K is convex, and n w(k) 2. Then, with high probability, the solution ẑ of the K-LASSO satisfies w(k) ẑ µz 0 2 n. Assume that K = µr DB d 1 z 0 = Dω 0 for some ω 0 R B d 1. Assume that the columns of D are normalized. Then w(k) R log(d). Martin Genzel Sparse Proteomics Analysis (SPA) WiCoS 2015 16 / 19

A Recovery Guarantee for SPA Theorem (G. 15) Suppose that s k N (0, I M ). Let z 0 S M 1 and assume that there exists R > 0 such that z 0 = Dω 0 for some ω 0 R B1 d. The observations follow y k = sign( s k, z 0 ) = sign( x k, ω 0 ), k = 1,..., n. and the number of samples satisfies n R 2 log(d). Then, with high probability, the solution of the LASSO n ẑ = argmin (y k s k, z ) 2 z R DB1 d k=1 satisfies ( ) 1/4 2 ẑ π z 0 2 R2 log(d) n. Martin Genzel Sparse Proteomics Analysis (SPA) WiCoS 2015 17 / 19

A Recovery Guarantee for SPA Theorem (G. 15) Suppose that s k N (0, I M ). Let z 0 S M 1 and assume that there exists R > 0 such that z 0 = Dω 0 for some ω 0 R B1 d. The observations follow y k = sign( s k, z 0 ) = sign( x k, ω 0 ), k = 1,..., n. and the number of samples satisfies n R 2 log(d). Then, with high probability, the solution of the LASSO n ẑ = D ˆω = D argmin (y k x k, ω ) 2 ω R B1 d k=1 satisfies ( ) 1/4 2 D ˆω π Dω 2 0 2 = ẑ π z 0 2 R2 log(d) n. Martin Genzel Sparse Proteomics Analysis (SPA) WiCoS 2015 17 / 19

Practical Relevance for MS-Data? Extensions: Baseline noise nk N (0, σ 2 I d ) Non-trivial covariance matrix, i.e., sk N (0, Σ) Adversarial bit-flips in the model y k = sign( x k, ω 0 ) How to achieve normalized columns in D? How to guarantee that R s, i.e., s-sparse vectors are allowed? Standardize the data (centering + normalizing) Given ˆω, how to switch over to the signal space? (D is unknown) Identify supp( ˆω) with peaks (manual approach) Martin Genzel Sparse Proteomics Analysis (SPA) WiCoS 2015 18 / 19

Practical Relevance for MS-Data? Extensions: Baseline noise nk N (0, σ 2 I d ) Non-trivial covariance matrix, i.e., sk N (0, Σ) Adversarial bit-flips in the model y k = sign( x k, ω 0 ) How to achieve normalized columns in D? How to guarantee that R s, i.e., s-sparse vectors are allowed? Standardize the data (centering + normalizing) Given ˆω, how to switch over to the signal space? (D is unknown) Identify supp( ˆω) with peaks (manual approach) Message of this talk An s-sparse disease fingerprint can be accurately recovered from only O(s log(d)) samples! Martin Genzel Sparse Proteomics Analysis (SPA) WiCoS 2015 18 / 19

THANK YOU FOR YOUR ATTENTION! Further Reading M. Genzel Sparse Proteomics Analysis: Toward a Mathematical Foundation of Feature Selection and Disease Classification. Master s Thesis, 2015. Y. Plan, R. Vershynin The generalized Lasso with non-linear observations. arxiv:1502.04071, 2015.

What to Do Next? Development of an abstract framework What kind of properties should the dictionary D have? Extension/generalization of the results More complicated models and algorithms Numerical verification of the theory Other examples from real-world applications Bio-informatics, neuro-imaging, astronomy, chemistry,... Dictionary learning / Factor analysis What can we learn about D? Martin Genzel Sparse Proteomics Analysis (SPA) WiCoS 2015 19 / 19