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Hierarchical approximations for data reduction and learning at multiple scales

  • * Corresponding author: Prashant Shekhar

    * Corresponding author: Prashant Shekhar 
This work was funded by the grants NSF1821311, NSF1645053, NSF1621853
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  • This paper describes a hierarchical learning strategy for generating sparse representations of multivariate datasets. The hierarchy arises from approximation spaces considered at successively finer scales. A detailed analysis of stability, convergence and behavior of error functionals associated with the approximations are presented, along with a well chosen set of applications. Results show the performance of the approach as a data reduction mechanism for both synthetic (univariate and multivariate) and a real dataset (geo-spatial). The sparse representation generated is shown to efficiently reconstruct data and minimize error in prediction. The approach is also shown to generalize well to unseen samples, extending its prospective application to statistical learning problems.

    Mathematics Subject Classification: Primary:68W25, 65D15;Secondary:65D12.

    Citation:

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  • Figure 4.  Scalewise performance of Algorithm 1 on TF2. The plots in the left column((a), (c), (e) and (g)) show the density of basis functions at each scale, selected while identifying the sparse representation. Rank here refers to the cardinality of basis function set $ B^s $. The right column ((b), (d), (f) and (h)) shows the corresponding scalewise reconstruction of the underlying function. The small dots show the samples from true function while star markers represent the sparse representation $ D^s_{sparse} $. The solid curve is the reconstruction from the sparse representation. The fraction in bracket (for example (11/200) for (b)) shows the number of data points chosen out of 200 as $ D^s_{sparse} $

    Figure 1.  Univariate {(a): Gramancy and Lee function - Test function 1 (TF1); (b): 1-D Schwefel function - TF2} and multivariate {(c): Dropwave function - TF3; (d): 2-D Schwefel function - TF4} test functions considered for studying the performance of Algorithm 1

    Figure 2.  Convergence behavior on the test functions measured in 2-norm prediction error on the observed data. Top row ((a) and (b)) shows the performance on univariate functions with bottom row for multivariate functions ((c) and (d)). Each of the plots also show the Critical scale ($ S_c $) and Convergence scale ($ S_a $) along with the $ \% $ of data sampled as the sparse representation ($ D^{s}_{sparse} $) at each scale

    Figure 3.  Convergence measured as the decay of upper bound (equation (45)) to the inner product in the native Hilbert space between approximation $ A_sf $ and the approximation error $ E^s $ for considered univariate ((a): TF1 and (b): TF2) and multivariate ((c): TF3 and (d): TF4) test functions

    Figure 5.  Scalewise performance of Algorithm 1 on TF3. Left column ((a), (c), (e) and (g)) shows the distribution of the sparse representation selected at multiple scales. Here we have shown the projection of the sparse representation (star markers) on the X-Y plain for ease of presentation. The right column ((b), (d), (f) and (h)) shows the corresponding reconstruction for the dropwave test function from the respective $ D^s_{sparse} $. Again the fraction in the header of plots (right column) shows the proportion of dataset chosen as $ D^s_{sparse} $

    Figure 6.  Confidence and Prediction Intervals for reconstruction from $ D^s_{sparse} $ from scale 0 to scale 5 for TF2. Along with the approximation produced at these scales (solid curve), the plots also show the sparse representation selected (star marked points) with the $ 95\% $ t-confidence interval (thinner dark shaded region) and $ 95\% $t-prediction interval (broader light shaded region)

    Figure 7.  (Ⅰ): Importance ranking for the sparse representation (important points) sampled at scale 0 for TF2 based on ordering governed by pivoted QR decomposition; (Ⅱ) Histogram of the top 3 most important points selected for TF2 (shown as a solid curve in each subplot). Here along the columns we have the increment in scale (abbreviated as S) and along the rows we have shown the histogram of the first, second and third most important point respectively (represented with abbreviation I for Importance)

    Figure 8.  Comparative behavior of the decay of $ 2 $-$ norm $ error for the Multiscale extension algorithm from [5] (represented as 'Berm') and Algorithm 1 (represented as 'Shek') for the four test function TF1, TF2, TF3 and TF4 in (a), (b), (c) and (d) respectively

    Figure 9.  Performance on unseen samples for TF1, TF2, TF3 and TF4 by Algorithm 1 ($ Shek $). Here plots (a), (c), (e) and (g) ((e) and (g) just show the projection on XY plane) show the distribution of the training and testing data (50/50 % split) with corresponding plots on the right hand side ((b), (d), (f) and (g)) showing the reconstructions obtained just using the training data. The header on ((b), (d), (f) and (h)) also show the corresponding testing error

    Figure 10.  Performance of Algorithm 4 from [5] ($ Berm $) on TF1, TF2, TF3 and TF4 with same experimental setup (same training and testing data as (a), (c), (e) and (g) in figure 9). Here again the corresponding testing error is displayed as a header with corresponding reconstructions from training data

    Figure 11.  Generalization analysis with repeated (100 times) random splitting of data into training and testing (again 50/50 split). Here we compare the mean and standard deviation of the Root Mean Square Error (RMSE) for prediction on testing data ($ D_{test} $) by Algorithm 4 from [5] ($ Berm $) and our Algorithm 1 ($ Shek $)

    Figure 12.  (a): Contour plot for Digital Elevation Model (DEM) data with 60m resolution; (b): Training DEM obtained by sampling the nodes at even indices (from DEM in (a)) resulting in a 120m resolution DEM

    Figure 13.  Scale dependent reconstructions for the DEM dataset in figure 12 (a). Here (a), (b) and (c) show the reconstructed contour plot for scales 0, 2 and 6 respectively. The header on these plots also show the the proportion of dataset selected as the sparse representation along with the $ \infty-norm $ of the reconstruction error

    Figure 14.  Comparison of 60m DEM reconstruction by $ Shek $ (a) and $ Berm $ (b) using the 120m DEM training data shown in figure 12 (b). The header shows the RMSE for prediction of height at the testing locations (testing RMSE for $ Shek $: 2.74 and testing RMSE for $ Berm $: 10.7)

    Algorithm 1 Hierarchical approach
    1: INPUT:
          Hyperparameters: $ [(TOL \geq 0, T>0, P > 1) \in \mathbb{R}^3] $
          Dataset: $ [(X \in \mathbb{R}^{n \times d}) \subset \Omega $, $ (f|_{X} \in \mathbb{R}^n)] $
          Prediction points: $ [(X_{*} \in \mathbb{R}^{n^{*} \times d}) \subset \Omega] $
    2: OUTPUT:
          Convergence Scale: $ [S_a \in \mathbb{N}] $
          Sparse Representation ($ D_{sparse}^{S_a} $): $ [(X_{S_a} \in \mathbb{R}^{\{l_{S_a} \times d \}}),(C_{S_a} \in \mathbb{R}^{l_{S_a}})] $
          Predictions: $ [P^* \in \mathbb{R}^{n^{*}}] $
    3: Initialize: $ s =0 $
    4: while $ TRUE $ do
    5:      Compute covariance kernel: [$ G_s $ on $ X $ with $ (\epsilon_s = T/P^s $)]
    6:      Compute numerical Rank: [$ l_s = rank(G_s) $]
    7:      Remove sampling Bias: [$ (W = AG_{s}) $ with $ A \in \mathbb{R}^{k \times n} $ and $ (a_{i,j} \sim N(0,1)) $]
    8:      Generate permutation information: [$ WP_R = QR $]
    9:      Produce bases at scale s: $ [B^{s} = (G_{s}P_R)[:,1:l_s] $, \quad \text{with}\ $ B^{s} \in \mathbb{R}^{n \times l_s}] $
    10:      Subset the sparse representation in $ X_s $
    11:      Compute projection coordinate: [$ C_{s} = {(B^{s})}^{\dagger} f|_X $; $ {B^s}^{\dagger} = ({B^s}^TB^s)^{-1}{B^s}^T $]
    12:      Generate approximation at s: [$ (A_sf)|_{X} = B^{s}C_s $]
    13:      If $ (||f|_X- (A_sf)|_{X}||_2 \leq TOL) $ : $ S_a = s $; $ Break $
    14:      Update scale: [$ s=s+1 $]
    15: end while
    16: Compute bases for prediction at $ X_{*} $: [$ G^{*}_{S_a} $ centered at $ X_{S_a} $with $ (\epsilon_{S_a} = T/P^{S_a} $)]
    17: Predict: $ P^* = G^{*}_{S_a} C_{S_a} $
     | Show Table
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    Table 1.  Comparison of training and prediction time for Algorithm 1 ($ Shek $) and Algorithm 4 from [5] ($ Berm $). For training analysis, we have considered the case of different training data sizes ($ |X| $) for all test functions. Correspondingly, for testing prediction latency, we again have sets $ |X_{pred}| $ of different cardinality. For each case, the approach taking shorter time has been boldfaced

    Training time (s) Prediction time (s)
    $ |X| $ $ Shek $ $ Berm $ $ |X_{pred}| $ $ Shek $ $ Berm $
    $ TF1 $ 50 4.61e-03 3.25e-02 100 1.47e-02 7.22e-02
    100 9.25e-03 1.40e-02 200 4.15e-02 3.09e-01
    150 1.60e-02 2.93e-02 300 6.21e-02 5.82e-01
    200 2.55e-02 5.63e-02 400 8.18e-02 1.08
    250 3.77e-02 9.29e-02 500 1.12e-01 1.81
    300 5.64e-02 1.25e-01 600 1.31e-01 2.38
    350 6.51e-02 1.93e-01 700 1.48e-01 3.56
    400 8.36e-02 2.21e-01 800 1.70e-01 4.15
    $ TF2 $ 50 5.68e-03 9.21e-03 100 2.17e-02 8.35e-02
    100 1.35e-02 1.49e-02 200 6.75e-02 3.00e-01
    150 2.86e-02 2.80e-02 300 1.53e-01 6.36e-01
    200 5.28e-02 5.27e-02 400 2.71e-01 1.07
    250 8.97e-02 8.94e-02 500 4.09e-01 1.81
    300 1.15e-01 1.23e-01 600 5.91e-01 2.29
    350 1.86e-01 1.75e-01 700 8.06e-01 3.52
    400 2.12e-01 2.17e-01 800 1.08 4.10
    $ TF3 $ 900 8.39e-01 7.98e-01 1225 3.13 8.67
    1600 4.19 6.40 2025 5.29 2.14e+01
    2500 2.76e+01 2.67e+01 3025 7.91 3.47e+01
    3600 7.19e+01 8.24e+01 4225 1.16e+01 4.95e+01
    4900 1.72e+02 1.99e+02 5625 1.59e+01 6.87e+01
    6400 3.84e+02 4.44e+02 7225 2.05e+01 8.73e+01
    8100 7.71e+02 8.88e+02 9025 2.75e+01 1.08e+02
    10000 1.44e+03 1.88e+03 11025 3.27e+01 2.32e+02
    $ TF4 $ 900 1.42 8.70e-01 1225 4.10 1.53e+01
    1600 6.94 8.68 2025 1.05e+01 3.21e+01
    2500 3.51e+01 3.44e+01 3025 2.25e+01 5.44e+01
    3600 1.27e+02 1.27e+02 4225 5.25e+01 1.32e+02
    4900 2.94e+02 2.95e+02 5625 8.78e+01 2.02e+02
    6400 7.90e+02 7.94e+02 7225 1.52e+02 3.96e+02
    8100 1.49e+03 1.49e+03 9025 2.24e+02 5.37e+02
    10000 2.69e+03 2.69e+03 11025 3.37e+02 6.90e+02
     | Show Table
    DownLoad: CSV
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