This paper presents a mathematical interpretation of self-attention by connecting it to distributional semantics principles. We show that self-attention emerges from projecting corpus-level co-occurrence statistics into sequence context. Starting from the co-occurrence matrix underlying GloVe embeddings, we demonstrate how the projection naturally captures contextual influence, with the query-key-value mechanism arising as the natural asymmetric extension for modeling directional relationships. Positional encodings and multi-head attention then follow as structured refinements of this same projection principle. Our analysis demonstrates that the Transformer architecture's particular algebraic form follows from these projection principles rather than being an arbitrary design choice.

The Apperception Engine is an unsupervised learning system. Given a sequence of sensory inputs, it constructs a symbolic causal theory that both explains the sensory sequence and also satisfies a set of unity conditions. The unity conditions insist that the constituents of the theory - objects, properties, and laws - must be integrated into a coherent whole. Once a theory has been constructed, it can be applied to predict future sensor readings, retrodict earlier readings, or impute missing readings. In this paper, we evaluate the Apperception Engine in a diverse variety of domains, including cellular automata, rhythms and simple nursery tunes, multi-modal binding problems, occlusion tasks, and sequence induction intelligence tests. In each domain, we test our engine's ability to predict future sensor values, retrodict earlier sensor values, and impute missing sensory data. The engine performs well in all these domains, significantly outperforming neural net baselines and state of the art inductive logic programming systems. These results are significant because neural nets typically struggle to solve the binding problem (where information from different modalities must somehow be combined together into different aspects of one unified object) and fail to solve occlusion tasks (in which objects are sometimes visible and sometimes obscured from view). We note in particular that in the sequence induction intelligence tests, our system achieved human-level performance. This is notable because our system is not a bespoke system designed specifically to solve intelligence tests, but a general-purpose system that was designed to make sense of any sensory sequence.

This paper attempts to answer a central question in unsupervised learning: what does it mean to "make sense" of a sensory sequence? In our formalization, making sense involves constructing a symbolic causal theory that explains the sensory sequence and satisfies a set of unity conditions. This model was inspired by Kant's discussion of the synthetic unity of apperception in the Critique of Pure Reason. On our account, making sense of sensory input is a type of program synthesis, but it is unsupervised program synthesis. Our second contribution is a computer implementation, the Apperception Engine, that was designed to satisfy the above requirements. Our system is able to produce interpretable human-readable causal theories from very small amounts of data, because of the strong inductive bias provided by the Kantian unity constraints. A causal theory produced by our system is able to predict future sensor readings, as well as retrodict earlier readings, and "impute" (fill in the blanks of) missing sensory readings, in any combination. We tested the engine in a diverse variety of domains, including cellular automata, rhythms and simple nursery tunes, multi-modal binding problems, occlusion tasks, and sequence induction IQ tests. In each domain, we test our engine's ability to predict future sensor values, retrodict earlier sensor values, and impute missing sensory data. The Apperception Engine performs well in all these domains, significantly out-performing neural net baselines. We note in particular that in the sequence induction IQ tasks, our system achieved human-level performance. This is notable because our system is not a bespoke system designed specifically to solve IQ tasks, but a general purpose apperception system that was designed to make sense of any sensory sequence.