Learning from Substitutable and Complementary Relations for Graph-based Sequential Product Recommendation

Sequential product recommendation, aiming at predicting the products that a target user will interact with soon, has become a hotspot topic. Most of the sequential recommendation models focus on learning from users’ interacted product sequences in a purely data-driven manner. However, they largely overlook the knowledgeable substitutable and complementary relations between products. To address this issue, we propose a novel Substitutable and Complementary Graph-based Sequential Product Recommendation model, namely, SCG-SPRe. The innovations of SCG-SPRe lie in its two main modules: (1) The module of interactive graph neural networks jointly encodes the high-order product correlations in the substitutable graph and the complementary graph into two types of relation-specific product representations. (2) The module of kernel-enhanced transformer networks adaptively fuses multiple temporal kernels to characterize the unique temporal patterns between a candidate product to be recommended and any interacted product in a target behavior sequence. Thanks to the seamless integration of the two modules, SCG-SPRe obtains candidate-dependent user representations for different candidate products to compute the corresponding ranking scores. We conduct extensive experiments on three public datasets, demonstrating SCG-SPRe is superior to competitive sequential recommendation baselines and validating the benefits of explicitly modeling the product-product relations.

On the Risks of Using the Sequential Product-Level SSNIP Approach to Identify Relevant Antitrust Markets‡

In a recent influential paper Coate et al. (2021) have criticized the sequential product-level approach to market definition in merger review. They argue that a simultaneous market-level approach to critical loss is more appropriate than a product-level critical loss analysis, because under certain plausible demand scenarios (nonlinear demand functions) the latter could yield the wrong answer on market definition—i.e., excessively broad or narrow markets. We extend their analysis by showing that a sequential product-level approach actually leads to an excessively narrow market definition when the typical nonlinear demand functions used in merger analysis are employed.

On the Decomposition of Generalized Semiautomata

Semiautomata are abstractions of electronic devices that are deterministic finite-state machines having inputs but no outputs. Generalized semiautomata are obtained from stochastic semiautomata by dropping the restrictions imposed by probability. It is well-known that each stochastic semiautomaton can be decomposed into a sequential product of a dependent source and deterministic semiautomaton making partly use of the celebrated theorem of Birkhoff-von Neumann. It will be shown that each generalized semiautomaton can be partitioned into a sequential product of a generalized dependent source and a deterministic semiautomaton.

The three types of normal sequential effect algebras

A sequential effect algebra (SEA) is an effect algebra equipped with a sequential product operation modeled after the Lüders product (a,b)↦aba on C∗-algebras. A SEA is called normal when it has all suprema of directed sets, and the sequential product interacts suitably with these suprema. The effects on a Hilbert space and the unit interval of a von Neumann or JBW algebra are examples of normal SEAs that are in addition convex, i.e. possess a suitable action of the real unit interval on the algebra. Complete Boolean algebras form normal SEAs too, which are convex only when 0=1.We show that any normal SEA E splits as a direct sum E=Eb⊕Ec⊕Eac of a complete Boolean algebra Eb, a convex normal SEA Ec, and a newly identified type of normal SEA Eac we dub purely almost-convex.Along the way we show, among other things, that a SEA which contains only idempotents must be a Boolean algebra; and we establish a spectral theorem using which we settle for the class of normal SEAs a problem of Gudder regarding the uniqueness of square roots. After establishing our main result, we propose a simple extra axiom for normal SEAs that excludes the seemingly pathological a-convex SEAs. We conclude the paper by a study of SEAs with an associative sequential product. We find that associativity forces normal SEAs satisfying our new axiom to be commutative, shedding light on the question of why the sequential product in quantum theory should be non-associative.