Findings of average differences between females and males in the structure of specific brain regions are often interpreted as indicating that the typical male brain is different from the typical female brain. An alternative interpretation is that the brain types typical of females are also typical of males, and sex differences exist only in the frequency of rare brain types. Here we contrasted the two hypotheses by analyzing the structure of 2176 human brains using three analytical approaches. An anomaly detection analysis showed that brains from females are almost as likely to be classified as “normal male brains,” as brains from males are, and vice versa. Unsupervised clustering algorithms revealed that common brain “types” are similarly common in females and in males and that a male and a female are almost as likely to have the same brain “type” as two females or two males are. Large sex differences were found only in the frequency of some rare brain “types.” Last, supervised clustering algorithms revealed that the brain “type(s)” typical of one sex category in one sample could be typical of the other sex category in another sample. The present findings demonstrate that even when similarity and difference are defined mathematically, ignoring biological or functional relevance, sex category (i.e., whether one is female or male), is not a major predictor of the variability of human brain structure. Rather, the brain types typical of females are also typical of males, and vice versa, and large sex differences are found only in the prevalence of some rare brain types. We discuss the implications of these findings to studies of the structure and function of the human brain.

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In this paper, a reduced dimensionality representation is learned from multiple views of the processed data. These multiple views can be obtained, for example, when the same underlying process is observed using several different modalities, or measured with different instrumentation. The goal is to effectively utilize the availability of such multiple views for various purposes such as non-linear embedding, manifold learning, spectral clustering, anomaly detection and non-linear system identification. The proposed method, which is called multi-view, exploits the intrinsic relation within each view as well as the mutual relations between views. This is achieved by defining a cross-view model in which an implied random walk process is restrained to hop between objects in the different views. This multi-view method is robust to scaling and it is insensitive to small structural changes in the data. Within this framework, new diffusion distances are defined to analyze the spectra of the implied kernels. The applicability of the multi-view approach is demonstrated for clustering, classification and manifold learning using both artificial and real data. 2 The problem of learning from two views has been studied in the field of spectral clustering. Most of these studies have been focused on classification and clustering that are based on spectral characteristics of the data while using two or more sampled views. Some approaches, which address this problem, are Bilinear Model [9], Partial Least Squares [10] and Canonical Correlation Analysis [11]. These methods are powerful for learning the relation between different views but do not provide separate insights or combined into the low dimensional geometry or structure of each view. Recently, a few kernel based methods (e.g [12]) propose a model of co-regularizing kernels in both views in a way that resembles joint diagonalization. It is done by searching for an orthogonal transformation that maximizes the diagonal terms of the kernel matrices obtained from all views. A penalty term, which incorporates the disagreement between clusters from the views, was added. Their algorithm is based on alternating maximization procedure. A mixture of Markov chains is proposed in [13] to model multiple views in order to apply spectral clustering. It deals with two cases in graph theory: directed and undirected graph where the second case is related to our work. This approach converges the undirected graph problem to a Markov chains averaging where each is constructed separately within the views. A way to incorporate a given multiple metrics for the same data using a cross diffusion process is described in [14]. They define a new diffusion distance which is useful for classification, clustering or retrieval tasks. However, the proposed process is not symmetrical thus does not allow to compute an embedding. An iterative algorithm for spectral clustering is proposed in [15]. The idea is to iteratively modify each view using the representation of the other view. The problem of two manifolds, ...

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Diffusion Maps (DM), and other kernel methods, are utilized for the analysis of high dimensional datasets. The DM method uses a Markovian diffusion process to model and analyze data. A spectral analysis of the DM kernel yields a map of the data into a low dimensional space, where Euclidean distances between the mapped data points represent the diffusion distances between the corresponding high dimensional data points. Many machine learning methods, which are based on the Euclidean metric, can be applied to the mapped data points in order to take advantage of the diffusion relations between them. However, a significant drawback of the DM is the need to apply spectral decomposition to a kernel matrix, which becomes infeasible for large datasets. In this paper, we present an efficient approximation of the DM embedding. The presented approximation algorithm produces a dictionary of data points by identifying a small set of informative representatives. Then, based on this dictionary, the entire dataset is efficiently embedded into a low dimensional space. The Euclidean distances in the resulting embedded space approximate the diffusion distances. The properties of the presented embedding and its relation to DM method are analyzed and demonstrated.

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