Recent works have derived neural networks with online correlation-based learning rules to perform \textit{kernel similarity matching}. These works applied existing linear similarity matching algorithms to nonlinear features generated with random Fourier methods. In this paper attempt to perform kernel similarity matching by directly learning the nonlinear features. Our algorithm proceeds by deriving and then minimizing an upper bound for the sum of squared errors between output and input kernel similarities. The construction of our upper bound leads to online correlation-based learning rules which can be implemented with a 1 layer recurrent neural network. In addition to generating high-dimensional linearly separable representations, we show that our upper bound naturally yields representations which are sparse and selective for specific input patterns. We compare the approximation quality of our method to neural random Fourier method and variants of the popular but non-biological ``Nystr{\"o}m'' method for approximating the kernel matrix. Our method appears to be comparable or better than randomly sampled Nystr{\"o}m methods when the outputs are relatively low dimensional (although still potentially higher dimensional than the inputs) but less faithful when the outputs are very high dimensional.

Sparse coding has been proposed as a theory of visual cortex and as an unsupervised algorithm for learning representations. We show empirically with the MNIST data set that sparse codes can be very sensitive to image distortions, a behavior that may hinder invariant object recognition. A locally linear analysis suggests that the sensitivity is due to the existence of linear combinations of active dictionary elements with high cancellation. A nearest-neighbor classifier is shown to perform worse on sparse codes than original images. For a linear classifier with a sufficiently large number of labeled examples, sparse codes are shown to yield higher accuracy than original images, but no higher than a representation computed by a random feedforward net. Sensitivity to distortions seems to be a basic property of sparse codes, and one should be aware of this property when applying sparse codes to invariant object recognition.

We show dense voxel embeddings learned via deep metric learning can be employed to produce a highly accurate segmentation of neurons from 3D electron microscopy images. A "metric graph" on a set of edges between voxels is constructed from the dense voxel embeddings generated by a convolutional network. Partitioning the metric graph with long-range edges as repulsive constraints yields an initial segmentation with high precision, with substantial accuracy gain for very thin objects. The convolutional embedding net is reused without any modification to agglomerate the systematic splits caused by complex "self-contact" motifs. Our proposed method achieves state-of-the-art accuracy on the challenging problem of 3D neuron reconstruction from the brain images acquired by serial section electron microscopy. Our alternative, object-centered representation could be more generally useful for other computational tasks in automated neural circuit reconstruction.