Recommendation algorithms can make a customer’s online shopping experience quicker and more efficient by suggesting complementary products whenever the shopper adds a product to their basket. Did the customer buy peanut butter? The algorithm recommends several brands of jelly to add next. 

These algorithms typically work by associating purchased items with items other shoppers have frequently purchased alongside them. If the shopper’s habits, tastes, or interests closely resemble those of previous customers, such recommendations might save time, jog the memory, and be a welcome addition to the shopping experience. 

But what if the shopper is buying peanut butter to stuff a dog toy or bait a mousetrap? What if the shopper prefers honey or bananas with their peanut butter? The recommendation algorithm will offer less useful suggestions, costing the retailer a sale and potentially annoying the customer. 

New research led by Negin Entezari, who recently received a doctoral degree in computer science at UC Riverside, Instacart collaborators, and her doctoral advisor Vagelis Papalexakis, brings a methodology called tensor decomposition—used by scientists to find patterns in massive volumes of data—into the world of commerce to recommend complementary products more carefully tailored to customer preferences. 

Tensors can be pictured as multi-dimensional cubes and are used to model and analyze data with many different components, called multi-aspect data. Data closely related to other data can be connected in a cube arrangement and related to other cubes to uncover patterns in the data. 

“Tensors can be used to represent customers’ shopping behaviors,” said Entezari. “Each mode of a 3-mode tensor can capture one aspect of a transaction. Customers form one mode of the tensor and the second and third mode captures product-to-product interactions by considering products co-purchased in a single transaction.”

For example, three hypothetical shoppers—A, B, and C— make the following purchases:

A: Buys hot dogs, hot dog buns, Coke, and mustard in one transaction.
B: Makes three separate transactions: Basket 1: Hot dogs and hot dog buns; Basket 2: Coke; Basket 3: Mustard
C: Hot dogs, hot dog buns, and mustard in one transaction.

To a conventional matrix-based algorithm, Customer A is identical to Customer B because they bought the same items. Using tensor decomposition, however, Customer A is more closely related to Customer C because their behavior was similar. Both had similar products co-purchased in a single transaction, even though their purchases differed slightly. 

The typical recommendation algorithm makes predictions based on the item the customer just purchased while tensor decomposition can make recommendations based on what is already in the user’s whole basket. Thus, if a shopper has dog food and peanut butter in their basket but no bread, a tensor-based recommendation algorithm might suggest a fillable dog chew toy instead of jelly if other users have also made that purchase.

“Tensors are multidimensional structures that allow modeling of complex, heterogeneous data,” said Papalexakis, an associate professor of computer science and engineering. “Instead of simply noticing which products are purchased together, there’s a third dimension. These products are purchased by this kind of user and the algorithm tries to determine which kinds of users are creating this match.”

To test their method, Entezari, Papalexakis, and co-authors Haixun Wang, Sharath Rao, and Shishir Kumar Prasad, all researchers for Instacart, used an Instacart public dataset to train their algorithm. They found that their method outperformed state-of-the-art methods for predicting customer-specific complementary product recommendations. Though more work is needed, the authors conclude that big data tensor decomposition could eventually find a home in big business as well.

“Tensor methods, even though very powerful tools, are still more popular in academic research as far as recommendation systems go,” said Papalexakis. “For the industry to adopt them we must demonstrate it’s worthwhile and relatively painless to substitute for whatever they have that works already.”

While previous research has shown the benefits of tensor modeling in recommendation problems, the new publication is the first to do so in the setting of complementary item recommendation, bringing tensor methods closer to industrial adoption and technology transfer in the context of recommendation systems.

“Tensor methods have been adopted successfully by industry before, with chemometrics and food quality being great examples, and every attempt like our work demonstrates the versatility of tensor methods in being able to tackle such a broad range of challenging problems in different domains,” said Papalexakis.

While for most of us cryptographic systems are things that just run “under the hood,” they are an essential element in the world of digital communications. However, the upcoming rise of quantum computers could shake the field of cryptography to its core. Fast algorithms running on these machines could break some of the most widely used cryptosystems, rendering them vulnerable. Well aware of this looming threat, cryptography researchers worldwide are working on novel encryption methods that can withstand attacks from quantum supercomputers. 

Chaos theory is actively being studied as a basis for post-quantum-era cryptosystems. In mathematics, chaos is a property of certain dynamic systems that makes them extremely sensitive to initial conditions. While technically deterministic (non-random), these systems evolve in such complex ways that predicting their long-term state with incomplete information is practically impossible, since even small rounding errors in the initial conditions yield diverging results. This unique characteristic of chaotic systems can be leveraged to produce highly secure cryptographic systems, as a team of researchers from Ritsumeikan University, Japan, showed in a recent study.

Led by Professor Takaya Miyano, the team developed an unprecedented stream cipher consisting of three cryptographic primitives based on independent mathematical models of chaos. The first primitive is a pseudorandom number generator based on the augmented Lorenz (AL) map. The pseudorandom numbers produced using this approach are used to create key streams for encrypting/decrypting messages, which take the stage in the second and perhaps most remarkable primitive—an innovative method for secret-key exchange.

This novel strategy for exchanging secret keys specifying the AL map is based on the synchronization of two chaotic Lorenz oscillators, which can be independently and randomly initialized by the two communicating users, without either of them knowing the state of the other’s oscillator. To conceal the internal states of these oscillators, the communicating users (the sender and the receiver) mask the value of one of the variables of their oscillator by multiplying it with a locally generated random number. The masked value of the sender is then sent to the receiver and vice-versa. After a short time, when these back-and-forth exchanges cause both oscillators to sync up almost perfectly to the same state despite the randomization of the variables, the users can mask and exchange secret keys and then locally unmask them with simple calculations.

Finally, the third primitive is a hash function based on the logistic map (a chaotic equation of motion), which allows the sender to send a hash value and, in turn, allows the receiver to ensure that the received secret key is correct, i.e., the chaotic oscillators were synchronized properly.

The researchers showed that a stream cipher assembled using these three primitives is extremely secure and resistant to statistical attacks and eavesdropping since it is mathematically impossible to synchronize their oscillator to either the sender’s or the receiver’s ones. This is an unprecedented achievement, as Prof. Miyano states: “Most chaos-based cryptosystems can be broken by attacks using classical computers within a practically short time. In contrast, our methods, especially the one for secret-key exchange, appear to be robust against such attacks and, more importantly, even hard to break using quantum computers.”

In addition to its security, the proposed key exchange method applies to existing block ciphers, such as the widely used Advanced Encryption Standard (AES). Moreover, the researchers could implement their chaos-based stream cipher on the Raspberry Pi 4, a small-scale computer, using Python 3.8. They even used it to securely transmit a famous painting by Johannes Vermeer between Kusatsu and Sendai, two places in Japan 600 km apart. “The implementation and running costs of our cryptosystem are remarkably low compared with those of quantum cryptography,” highlights Prof. Miyano, “Our work thus provides a cryptographic approach that guarantees the privacy of daily communications between people all over the world in the post-quantum era.”

With such power of chaos-based cryptography, we may not have much to worry about the dark sides of quantum computing.