# A Generative Model for Deep Convolutional Learning

###### Abstract

A generative model is developed for deep (multi-layered) convolutional dictionary learning. A novel probabilistic pooling operation is integrated into the deep model, yielding efficient bottom-up (pretraining) and top-down (refinement) probabilistic learning. Experimental results demonstrate powerful capabilities of the model to learn multi-layer features from images, and excellent classification results are obtained on the MNIST and Caltech 101 datasets.

A Generative Model for Deep Convolutional Learning

Yunchen Pu, Xin Yuan and Lawrence Carin |
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Department of Electrical and Computer Engineering, Duke University, Durham, NC, 27708, USA |

{yunchen.pu,xin.yuan,lcarin}@duke.edu |

## 1 Introduction

We develop a deep generative statistical model, which starts at the highest-level features, and maps these through a sequence of layers, until ultimately mapping to the data plane (e.g., an image). The feature at a given layer is mapped via a multinomial distribution to one feature in a block of features at the layer below (and all other features in the block at the next layer are set to zero). This is analogous to the method in Lee et al. (2009), in the sense of imposing that there is at most one non-zero activation within a pooling block. We use bottom-up pretraining, in which initially we sequentially learn parameters of each layer one at a time, from bottom to top, based on the features at the layer below. However, in the refinement phase, all model parameters are learned jointly, top-down. Each consecutive layer in the model is locally conjugate in a statistical sense, so learning model parameters may be readily performed using sampling or variational methods.

## 2 Modeling Framework

Assume gray-scale images , with ; the images are analyzed jointly to learn the convolutional dictionary . Specifically consider the model

(1) |

where is the convolution operator, denotes the Hadamard (element-wise) product, the elements of are in , the elements of are real, and represents the residual. indicates which shifted version of is used to represent .

Assume an -layer model, with layer the top layer, and layer 1 at the bottom, closest to the data. In the pretraining stage, the output of layer is the input to layer , after pooling. Layer has dictionary elements, and we have:

(2) | |||||

(3) |

The expression may be viewed as a 3D entity, with its -th plane defined by a “pooled” version of .

The 2D activation map is partitioned into dimensional contiguous blocks (pooling blocks with respect to layer of the model); see the left part of Figure 1. Associated with each block of pixels in is one pixel at layer of ; the relative locations of the pixels in are the same as the relative locations of the blocks in . Within each block of , either all pixels are zero, or only one pixel is non-zero, with the position of that pixel selected stochastically via a multinomial distribution. Each pixel at layer of equals the largest-amplitude element in the associated block of (, max pooling).

The learning performed with the top-down generative model (right part of Fig. 1) constitutes a refinement of the parameters learned during pretraining, and the excellent initialization constituted by the parameters learned during pretraining is key to the subsequent model performance.

## 3 Experimental Results

We here apply our model to the MNIST and Caltech 101 datasets.

### MNIST Dataset

Methods | Test error |
---|---|

0.35% | |

MCDNN Ciresan et al. (2012) | 0.23% |

SPCNN Zeiler & Fergus (2013) | 0.47% |

0.89% | |

Ours, 2-layer model + 1-layer features | 0.42% |

Table 1 summaries the classification results of our model compared with some related results, on the MNIST data. The second (top) layer features corresponding to the refined dictionary are sent to a nonlinear support vector machine (SVM) (Chang & Lin, 2011) with Gaussian kernel, in a one-vs-all multi-class classifier, with classifier parameters tuned via 5-fold cross-validation (no tuning on the deep feature learning).

### Caltech 101 Dataset

# Training Images per Category | 15 | 30 |
---|---|---|

DN Zeiler et al. (2010) | 58.6 % | 66.9% |

CBDN Lee et al. (2009) | 57.7 % | 65.4% |

HBP Chen et al. (2013) | 58% | 65.7% |

ScSPM Yang et al. (2009) | 67 % | 73.2% |

P-FV Seidenari et al. (2014) | 71.47% | 80.13% |

R-KSVD Li et al. (2013) | 79 % | 83% |

Convnet Zeiler & Fergus (2014) | 83.8 % | 86.5% |

Ours, 2-layer model + 1-layer features | 70.02% | 80.31% |

Ours, 3-layer model + 1-layer features | 75.24% | 82.78% |

We next consider the Caltech 101 dataset.For Caltech 101 classification, we follow the setup in Yang et al. (2009), selecting 15 and 30 images per category for training, and testing on the rest. The features of testing images are inferred based on the top-layer dictionaries and sent to a multi-class SVM; we again use a Gaussian kernel non-linear SVM with parameters tuned via cross-validation. Ours and related results are summarized in Table 2.

## 4 Conclusions

A deep generative convolutional dictionary-learning model has been developed within a Bayesian setting. The proposed framework enjoys efficient bottom-up and top-down probabilistic inference. A probabilistic pooling module has been integrated into the model, a key component to developing a principled top-down generative model, with efficient learning and inference. Extensive experimental results demonstrate the efficacy of the model to learn multi-layered features from images.

## References

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