Ebrahim Ardeshir-Larijani

Sajjad Hashemian, Mohammad Saeed Arvenaghi, Ebrahim Ardeshir-Larijani

In this paper, we introduce new algorithms for Principal Component Analysis (PCA) with outliers. Utilizing techniques from computational geometry, specifically higher-degree Voronoi diagrams, we navigate to the optimal subspace for PCA even in the presence of outliers. This approach achieves an optimal solution with a time complexity of $n^{d+\mathcal{O}(1)}\text{poly}(n,d)$. Additionally, we present a randomized algorithm with a complexity of $2^{\mathcal{O}(r(d-r))} \times \text{poly}(n, d)$. This algorithm samples subspaces characterized in terms of a Grassmannian manifold. By employing such sampling method, we ensure a high likelihood of capturing the optimal subspace, with the success probability $(1 - δ)^T$. Where $δ$ represents the probability that a sampled subspace does not contain the optimal solution, and $T$ is the number of subspaces sampled, proportional to $2^{r(d-r)}$. Our use of higher-degree Voronoi diagrams and Grassmannian based sampling offers a clearer conceptual pathway and practical advantages, particularly in handling large datasets or higher-dimensional settings.

Herbod Pourali, Sajjad Hashemian, Ebrahim Ardeshir-Larijani

We introduce an expander-sketching framework for list-decodable linear regression that achieves sample complexity $\tilde{O}((d+\log(1/δ))/α)$, list size $O(1/α)$, and near input-sparsity running time $\tilde{O}(\mathrm{nnz}(X)+d^{3}/α)$ under standard sub-Gaussian assumptions. Our method uses lossless expanders to synthesize lightly contaminated batches, enabling robust aggregation and a short spectral filtering stage that matches the best known efficient guarantees while avoiding SoS machinery and explicit batch structure.

Amir Kermanshahani, Ebrahim Ardeshir-Larijani, Rakesh Saini et al.

Quantum computing, with its ability to do exponentially faster computation compared to classical systems, has found novel applications in various fields such as machine learning and recommendation systems. Quantum Machine Learning (QML), which integrates quantum computing with machine learning techniques, presents powerful new tools for data processing and pattern recognition. This paper proposes a hybrid recommendation system that combines Quantum Hopfield Associative Memory (QHAM) with deep neural networks to improve the extraction and classification on the MovieLens 1M dataset. User archetypes are clustered into multiple unique groups using the K-Means algorithm and converted into polar patterns through the encoder's activation function. These polar patterns are then integrated into the variational QHAM-based hybrid recommendation model. The system was trained using the MSE loss over 35 epochs in an ideal environment, achieving an ROC value of 0.9795, an accuracy of 0.8841, and an F-1 Score of 0.8786. Trained with the same number of epochs in a noisy environment using a custom Qiskit AER noise model incorporating bit-flip and readout errors with the same probabilities as in real quantum hardware, it achieves an ROC of 0.9177, an accuracy of 0.8013, and an F-1 Score equal to 0.7866, demonstrating consistent performance. Additionally, we were able to optimize the qubit overhead present in previous QHAM architectures by efficiently updating only one random targeted qubit. This research presents a novel framework that combines variational quantum computing with deep learning, capable of dealing with real-world datasets with comparable performance compared to purely classical counterparts. Additionally, the model can perform similarly well in noisy configurations, showcasing a steady performance and proposing a promising direction for future usage in recommendation systems.