We present a new probabilistic model of compact commutative Lie groups that produces invariant-equivariant and disentangled representations of data. To define the notion of disentangling, we borrow a fundamental principle from physics that is used to derive the elementary particles of a system from its symmetries. Our model employs a newfound Bayesian conjugacy relation that enables fully tractable probabilistic inference over compact commutative Lie groups -- a class that includes the groups that describe the rotation and cyclic translation of images. We train the model on pairs of transformed image patches, and show that the learned invariant representation is highly effective for classification.
Other, non-group-theoretical approaches to learning transformations and invariant representations exist @cite_8 . These were found to perform a kind of joint eigenspace analysis @cite_0 , which is somewhat similar to the irreducible reduction of a toroidal group.
@cite_0 @cite_8
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Representations that are invariant to translation, scale and other transformations, can considerably reduce the sample complexity of learning, allowing recognition of new object classes from very few examples|a hallmark of human recognition. Empirical estimates of one-dimensional projections of the distribution induced by a group of ane transformations are proven to represent a unique and invariant signature associated with an image. We show how projections yielding invariant signatures for future images can be learned automatically, and updated continuously, during unsupervised visual experience. A module performing ltering and pooling, like simple and complex cells as proposed by Hubel and Wiesel, can compute such estimates. Under this view, a pooling stage estimates a one-dimensional probability distribution. Invariance from observations through a restricted window is equivalent to a sparsity property w.r.t. to a transformation, which yields templates that are a) Gabor for optimal simultaneous invariance to translation and scale or b) very specic for complex, class-dependent transformations such as rotation in depth of faces. Hierarchical architectures consisting of this basic Hubel-Wiesel module inherit its properties of invariance, stability, and discriminability while capturing the compositional organization of the visual world in terms of wholes and parts, and are invariant to complex transformations that may only be locally ane. The theory applies to several existing deep learning convolutional architectures for image and speech recognition. It also suggests that the main computational goal of the ventral stream of visual cortex is to provide a hierarchical representation of new objects images which is invariant to transformations, stable, and discriminative for recognition|and that this representation may be continuously learned in an unsupervised way during development and natural visual experience. Learning invariant representations is an important problem in machine learning and pattern recognition. In this paper, we present a novel framework of transformation-invariant feature learning by incorporating linear transformations into the feature learning algorithms. For example, we present the transformation-invariant restricted Boltzmann machine that compactly represents data by its weights and their transformations, which achieves invariance of the feature representation via probabilistic max pooling. In addition, we show that our transformation-invariant feature learning framework can also be extended to other unsupervised learning methods, such as autoencoders or sparse coding. We evaluate our method on several image classification benchmark datasets, such as MNIST variations, CIFAR-10, and STL-10, and show competitive or superior classification performance when compared to the state-of-the-art. Furthermore, our method achieves state-of-the-art performance on phone classification tasks with the TIMIT dataset, which demonstrates wide applicability of our proposed algorithms to other domains. Learning invariant representations is an important problem in machine learning and pattern recognition. In this paper, we present a novel framework of transformation-invariant feature learning by incorporating linear transformations into the feature learning algorithms. For example, we present the transformation-invariant restricted Boltzmann machine that compactly represents data by its weights and their transformations, which achieves invariance of the feature representation via probabilistic max pooling. In addition, we show that our transformation-invariant feature learning framework can also be extended to other unsupervised learning methods, such as autoencoders or sparse coding. We evaluate our method on several image classification benchmark datasets, such as MNIST variations, CIFAR-10, and STL-10, and show competitive or superior classification performance when compared to the state-of-the-art. Furthermore, our method achieves state-of-the-art performance on phone classification tasks with the TIMIT dataset, which demonstrates wide applicability of our proposed algorithms to other domains. Despite the importance of image representations such as histograms of oriented gradients and deep Convolutional Neural Networks (CNN), our theoretical understanding of them remains limited. Aiming at filling this gap, we investigate three key mathematical properties of representations: equivariance, invariance, and equivalence. Equivariance studies how transformations of the input image are encoded by the representation, invariance being a special case where a transformation has no effect. Equivalence studies whether two representations, for example two different parametrisations of a CNN, capture the same visual information or not. A number of methods to establish these properties empirically are proposed, including introducing transformation and stitching layers in CNNs. These methods are then applied to popular representations to reveal insightful aspects of their structure, including clarifying at which layers in a CNN certain geometric invariances are achieved. While the focus of the paper is theoretical, direct applications to structured-output regression are demonstrated too.