Large knowledge bases (KBs) are useful for many AI tasks, but are difficult to integrate into modern gradient-based learning systems. Here we describe a framework for accessing soft symbolic database using only differentiable operators. For example, this framework makes it easy to conveniently write neural models that adjust confidences associated with facts in a soft KB; incorporate prior knowledge in the form of hand-coded KB access rules; or learn to instantiate query templates using information extracted from text. NQL can work well with KBs with millions of tuples and hundreds of thousands of entities on a single GPU.
NQL is closely related to TensorLog @cite_6 , a deductive database formalism which also can be compiled to Tensorflow. In fact, NQL was designed so that every expression in the target sublanguage used by TensorLog can be concisely and readably written in NQL. TensorLog, in turn, has semantics derived from other proof-counting'' logics such as stochastic logic programs (SLP) @cite_0 . TensorLog is also closely related to other differentiable first-order logics such as the differentiable theorem prover (DTP) @cite_4 , in which a proof for an example is unrolled into a network. DPT includes representation-learning as a component, as well as a template-instantiation approach similar to the one used in NQL. TensorLog and NQL are more restricted than DPT but also more scaleable: the current version of NQL can work well with KBs with millions of tuples and hundreds of thousands of entities, even on a single GPU.
@cite_0 @cite_4 @cite_6
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We present an implementation of a probabilistic first-order logic called TensorLog, in which classes of logical queries are compiled into differentiable functions in a neural-network infrastructure such as Tensorflow or Theano. This leads to a close integration of probabilistic logical reasoning with deep-learning infrastructure: in particular, it enables high-performance deep learning frameworks to be used for tuning the parameters of a probabilistic logic. Experimental results show that TensorLog scales to problems involving hundreds of thousands of knowledge-base triples and tens of thousands of examples. Many problems in computational neuroscience, neuroinformatics, pattern image recognition, signal processing and machine learning generate massive amounts of multidimensional data with multiple aspects and high dimensionality. Tensors (i.e., multi-way arrays) provide often a natural and compact representation for such massive multidimensional data via suitable low-rank approximations. Big data analytics require novel technologies to efficiently process huge datasets within tolerable elapsed times. Such a new emerging technology for multidimensional big data is a multiway analysis via tensor networks (TNs) and tensor decompositions (TDs) which represent tensors by sets of factor (component) matrices and lower-order (core) tensors. Dynamic tensor analysis allows us to discover meaningful hidden structures of complex data and to perform generalizations by capturing multi-linear and multi-aspect relationships. We will discuss some fundamental TN models, their mathematical and graphical descriptions and associated learning algorithms for large-scale TDs and TNs, with many potential applications including: Anomaly detection, feature extraction, classification, cluster analysis, data fusion and integration, pattern recognition, predictive modeling, regression, time series analysis and multiway component analysis. Keywords: Large-scale HOSVD, Tensor decompositions, CPD, Tucker models, Hierarchical Tucker (HT) decomposition, low-rank tensor approximations (LRA), Tensorization Quantization, tensor train (TT QTT) - Matrix Product States (MPS), Matrix Product Operator (MPO), DMRG, Strong Kronecker Product (SKP). Multi-dimensional arrays, or tensors, are increasingly found in fields such as signal processing and recommender systems. Real-world tensors can be enormous in size and often very sparse. There is a need for efficient, high-performance tools capable of processing the massive sparse tensors of today and the future. This paper introduces SPLATT, a C library with shared-memory parallelism for three-mode tensors. SPLATT contains algorithmic improvements over competing state of the art tools for sparse tensor factorization. SPLATT has a fast, parallel method of multiplying a matricide tensor by a Khatri-Rao product, which is a key kernel in tensor factorization methods. SPLATT uses a novel data structure that exploits the sparsity patterns of tensors. This data structure has a small memory footprint similar to competing methods and allows for the computational improvements featured in our work. We also present a method of finding cache-friendly reordering and utilizing them with a novel form of cache tiling. To our knowledge, this is the first work to investigate reordering and cache tiling in this context. SPLATT averages almost 30x speedup compared to our baseline when using 16 threads and reaches over 80x speedup on NELL-2. We introduce neural networks for end-to-end differentiable proving of queries to knowledge bases by operating on dense vector representations of symbols. These neural networks are constructed recursively by taking inspiration from the backward chaining algorithm as used in Prolog. Specifically, we replace symbolic unification with a differentiable computation on vector representations of symbols using a radial basis function kernel, thereby combining symbolic reasoning with learning subsymbolic vector representations. By using gradient descent, the resulting neural network can be trained to infer facts from a given incomplete knowledge base. It learns to (i) place representations of similar symbols in close proximity in a vector space, (ii) make use of such similarities to prove queries, (iii) induce logical rules, and (iv) use provided and induced logical rules for multi-hop reasoning. We demonstrate that this architecture outperforms ComplEx, a state-of-the-art neural link prediction model, on three out of four benchmark knowledge bases while at the same time inducing interpretable function-free first-order logic rules.