r20 - 25 Aug 2008 - 18:54:12 - DumitruErhanYou are here: TWiki >  Public Web  > BackgroundCorrelation

Impact of background pixel correlation

In order to explore the space of learning problems standing between mnist-back-rand and mnist-back-images, we set up an experiment where we could vary the amount of background pixel correlation. We are hence assuming that background correlation is the main characteristic that distinguishes mnist-back-images from mnist-back-random.

Correlated pixel noise was sampled from a zero-mean multivariate Gaussian distribution of dimension equal to the number of pixels: $s \sim \mathcal{N}(0,\Sigma)$. The covariance matrix, $\Sigma$, was defined as a convex combination of an identity matrix and a Gaussian kernel function. Representing the position of the $i$ th pixel with the vector $(x_{i}, y_{i})$, we have:

$\Sigma_{ij} = \gamma I_{x_{i} = x_{j}} I_{y_{i} = y_{j}} + (1-\gamma)e^{-\left(\frac{(x_{i}-x_{j})^{2}}{\sigma^{2}} + \frac{(y_{i}-y_{j})^{2}}{\sigma^2}\right)}$

with kernel bandwidth $\sigma = 6$. The Gaussian kernel induced a neighborhood correlation structure among pixels such that nearby pixels are more correlated that pixels further apart. For each sample from $\mathcal{N}(0,\Sigma)$, the pixel values $p$ (ranging from 0 to 1) were determined by passing elements of $s$ through an error function:

$p_{i} = \frac{2}{\pi} \int_{0}^{s_{i}/\sqrt{2}} e^{-t^2}\ dt $

We generated six datasets with varying degrees of neighborhood correlation by setting the mixture weight $\gamma$ to the values $\{0, 0.2, 0.4, 0.6, 0.8, 1\}$. The marginal distributions for each pixel $p_{i}$ is uniform(0,1) for each value of $\gamma$.

noise_variations.png

Download links

All the datasets are provided as zip archives. Each archive contains two files -- a training (and validation) set and a test set. We used the last 2000 examples of the training sets as validation sets in all cases but for rectangles (200) and, in the case of SVMs, retrained the models with the entire set after choosing the optimal parameters on these validation sets. Data is stored at one example per row, the features being space-separated. There are 784 features per example (=28*28 images), corresponding to the first 784 columns of each row. The last column is the label, which is 0 to 9 for the MNIST variations and 1 or 0 for the rectangles, rectangles-images and convex datasets.

Link to dataset Size File description
MNIST noise variation sets 304M packed; 1008M unpacked 6x12000 train, 6x2000 test

Contact

Please contact us if you are interested in the (MATLAB) scripts that we used to generate this data.

-- DumitruErhan - 21 Jun 2007

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