of Computer Science University of Colorado, Boulder activesubspaces.org! @DrPaulynomial! SLIDES AVAILABLE UPON REQUEST DISCLAIMER: These slides are meant to complement the oral presentation. Use out of context at your own risk. Thanks to: Jeff Hokanson (CU Boulder) Andrew Glaws (CU Boulder, NREL)
Several independent inputs Deterministic Continuous inputs / outputs “Smoothness” Z f( x ) d x APPROXIMATION OPTIMIZATION INTEGRATION ˜ f( x ) ⇡ f( x ) minimize x f( x ) TO DO:
we can do to data only in terms of some set of very restrictive assumptions under which that thing is best possible--- assumptions we know we CANNOT check in practice.”
the dimension. But what is dimension reduction? • dimensional analysis [Barrenblatt (1996)] • correlation-based reduction [Jolliffe (2002)] • sensitivity analysis [Saltelli et al. (2008)]
et al. (2015), Pinkus (2015), Diaconis and Shahshahani (1984), Donoho and Johnstone (1989) Compressed sensing: Fornasier et al. (2012), Cohen et al. (2012), Tyagi and Cevher (2014) Statistical regression: Friedman and Stuetzle (1981), Ichimura (1993), Hristache et al. (2001), Xia et al. (2002) Uncertainty quantification & computational science: Tipireddy and Ghanem (2014); Lei et al. (2015); Stoyanov and Webster (2015); Tripathy, Bilionis, and Gonzalez (2016); Li, Lin, and Li (2016); … f( x ) ⇡ g(UT x )
0.05 0.1 0.15 0.2 0.25 Active Variable 1 P max (watts) Constantine, Zaharatos, and Campanelli (2015) Evidence of 1d ridge structures across science and engineering models
5 10 15 f(x) Average velocity Glaws, Constantine, Shadid, and Wildey (2017) -1 0 1 wT 1 x 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 f(x) Induce magnetic field Evidence of 1d ridge structures across science and engineering models
3.70 Voltage [V] Constantine and Doostan (2017) 2 0 2 wT x 2.0 2.2 Capacity [mAh·cm 2] Evidence of 1d ridge structures across science and engineering models
1 2 3 4 5 -4 -2 0 2 4 Quantity of interest #10-4 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 Long length scale Short length scale Constantine, Dow, and Wang (2014) r · (aru) = 1, s 2 D u = 0, s 2 1 n · aru = 0, s 2 2 Input field Solution Short corr Long corr Evidence of 1d ridge structures across science and engineering models
x ) ⇡ g(UT x ) Use the conditional average: subspace coordinates What is g? complement subspace and coordinates conditional density Constantine, Dow, and Wang (2014); Constantine, Eftekhari, Hokanson, and Ward (2017) Ridge approximations is the best approximation [Pinkus (2015)] µ(UT x ) <latexit 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x ) d x = W ⇤W T Define the active subspace The average outer product of the gradient and its eigendecomposition, Partition the eigendecomposition, Rotate and separate the coordinates, ⇤ = ⇤1 ⇤2 , W = ⇥ W 1 W 2 ⇤ , W 1 2 Rm⇥n x = W W T x = W 1W T 1 x + W 2W T 2 x = W 1y + W 2z active variables inactive variables f = f( x ), x 2 Rm, rf( x ) 2 Rm, ⇢ : Rm ! R + Constantine, Dow, and Wang (2014) Some relevant literature Statistical regression: Samarov (1993), Hristache et al. (2001) Machine learning: Mukerjee, Wu, and Xiao (2010); Fukumizu and Leng (2014) Detection and estimation theory: van Trees (2001) The function, its gradient vector, and a given weight function:
x ) d x = W ⇤W T Define the active subspace The function, its gradient vector, and a given weight function: The average outer product of the gradient and its eigendecomposition: f = f( x ), x 2 Rm, rf( x ) 2 Rm, ⇢ : Rm ! R + Constantine, Dow, and Wang (2014) i = Z w T i rf( x ) 2 ⇢( x ) d x , i = 1, . . . , m average, squared, directional derivative along eigenvector eigenvalue Eigenvalues measure ridge structure with eigenvectors:
(3) Approximate with Monte Carlo, and compute eigendecomposition Equivalent to SVD of samples of the gradient Called an active subspace method in T. Russi’s 2010 Ph.D. thesis, Uncertainty Quantification with Experimental Data in Complex System Models C ⇡ 1 N N X j=1 rfj rfT j = ˆ W ˆ ⇤ ˆ W T 1 p N ⇥ rf1 · · · rfN ⇤ = ˆ W p ˆ ⇤ ˆ V T rfj = rf( xj) Constantine, Dow, and Wang (2014), Constantine and Gleich (2015, arXiv) xj ⇠ ⇢( x ) Estimate the active subspace with Monte Carlo
⇡ ˆ W 1 q ˆ ⇤1 ˆ V T 1 Low-rank approximation of the collection of gradients: Low-dimensional linear approximation of the gradient: f( x ) ⇡ g ⇣ ˆ W T 1 x ⌘ Approximate a function of many variables by a function of a few linear combinations of the variables: ✔ ✖ ✖ Remember the problem to solve span ( ˆ W 1) ⇡ { rf( x ) : x 2 supp ⇢( x ) }
m ) ◆ = ) | k ˆk | k " How many gradient samples? number of samples eigenvalue error (w.h.p.) subspace error (w.h.p.) Constantine and Gleich (2015) via Gittens and Tropp (2011), Stewart (1973) N = ⌦ ✓ L2 1"2 log( m ) ◆ = ) dist( W 1, ˆ W 1) 4 1" n n+1 bound on gradient dimension number of samples bound on gradient dimension
Recall the subspace error: Effect of estimated eigenvectors? Constantine, Dow, and Wang (2014) " = dist( W 1, ˆ W 1) / O (eigval. error) n n+1 f( x ) µ( ˆ W T 1 x ) L2(⇢) C ⇣ " ( 1 + · · · + n)1 2 + ( n+1 + · · · + m)1 2 ⌘
Define the error function: R(U) = 1 2 Z (f( x ) µ(UT x ))2 ⇢( x ) d x Minimize the error: minimize U R ( U ) subject to U 2 G ( n, m ) Grassmann manifold Constantine, Eftekhari, Hokanson, and Ward (2017) Ridge approximations best approximation
(m n)1 2 ⌘ ( n+1 + · · · + m)1 2 The active subspace is nearly stationary Assume (1) Lipschitz continuous function (2) Gaussian density function gradient on the Grassmann manifold active subspace Frobenius norm dimensions Lipschitz constant eigenvalues associated with inactive subspace Constantine, Eftekhari, Hokanson, and Ward (2017)
Minimize the misfit Minimize over polynomials and subspaces xj ⇠ ⇢( x ) Estimate the optimal subspace with discrete least squares minimize g2P p(Rn) U2G(n,m) N X j=1 ⇣ fj g(UT xj) ⌘2 Constantine, Eftekhari, Hokanson, and Ward (2017), Hokanson and Constantine (2018)
Constantine (2018) Variable projection Given subspace, fit polynomial Given polynomial coefficients, minimize over subspace Repeat Use pseudoinverse of Vandermonde matrix to express optimal polynomial coefficients Compute the derivative of the pseudoinverse of the Vandermonde matrix [Golub & Pereyra (1973)] on the Grassmann manifold [Edelman et al. (1998)] Run Newton on loss function Two contenders for the least squares problem minimize g2P p(Rn) U2G(n,m) N X j=1 ⇣ fj g(UT xj) ⌘2
Details under the hood How do you choose points? How do you choose the subspace dimension? What is your polynomial basis for the ridge approximation? Grassmann gradient of pseudo-inverse is complicated but has some nice simplifications Open questions Can you use other bases (frames, RBFs, …) ? Is the continuous optimization problem well-posed? How does the discrete optimization relate to the continuous optimization?
(2) Insights + which variables are important (3) Discoverable / checkable + eigenvalues + non-residual metrics: + plots in 1 and 2d E[ Var[ f | UT x ] ] SUMMARY :: Why I like ridge structure
to reduce the dimension! There are many notions of important subspaces; they arise in several applications. Important subspaces are discoverable and exploitable for answering science questions. TAKE HOMES
trade-off between discovering the low- dimensional structure vs. solving the original problem? Why are these structures so pervasive? What if my model doesn’t fit your setup? (no gradients, multiple outputs, correlated inputs, …) PAUL CONSTANTINE Assistant Professor University of Colorado Boulder activesubspaces.org! @DrPaulynomial! QUESTIONS? Active Subspaces SIAM (2015)