arXiv subjects

Tin T. Tran

Publications and source records attributed to Tin T. Tran.

16 recordsLinked to original sources

arXiv2026

On phase and norm retrieval by subspaces

This paper studies phase and norm retrieval by subspaces. We first investigate norm retrieval by hyperplanes. We show that if $N$ hyperplanes $\{φ_i^\perp\}_{i=1}^N\subset \mathbb{R}^N$ allow norm retrieval and the vectors $\{φ_i\}_{i=1}^N$ are linearly independent, then these vectors must be an orthonormal basis for $\mathbb{R}^N$. We then present several new properties of subspaces that allow phase and norm retrieval. In particular, we provide a complete classification of two proper subspaces that perform norm retrieval. It is known that the collection of norm-retrievable frames $\{φ_i\}_{i=1}^M$ in $\mathbb{R}^N$ is not dense in the set of all $M$-element frames when $M < 2N-1$. We extend this result to subspaces. Several alternative proofs of fundamental results in phase and norm retrieval are also provided.

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arXiv2025

SEP-GCN: Leveraging Similar Edge Pairs with Temporal and Spatial Contexts for Location-Based Recommender Systems

Recommender systems play a crucial role in enabling personalized content delivery amidst the challenges of information overload and human mobility. Although conventional methods often rely on interaction matrices or graph-based retrieval, recent approaches have sought to exploit contextual signals such as time and location. However, most existing models focus on node-level representation or isolated edge attributes, underutilizing the relational structure between interactions. We propose SEP-GCN, a novel graph-based recommendation framework that learns from pairs of contextually similar interaction edges, each representing a user-item check-in event. By identifying edge pairs that occur within similar temporal windows or geographic proximity, SEP-GCN augments the user-item graph with contextual similarity links. These links bridge distant but semantically related interactions, enabling improved long-range information propagation. The enriched graph is processed via an edge-aware convolutional mechanism that integrates contextual similarity into the message-passing process. This allows SEP-GCN to model user preferences more accurately and robustly, especially in sparse or dynamic environments. Experiments on benchmark data sets show that SEP-GCN consistently outperforms strong baselines in both predictive accuracy and robustness.

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arXiv2025

Combining social relations and interaction data in Recommender System with Graph Convolution Collaborative Filtering

A recommender system is an important subject in the field of data mining, where the item rating information from users is exploited and processed to make suitable recommendations with all other users. The recommender system creates convenience for e-commerce users and stimulates the consumption of items that are suitable for users. In addition to e-commerce, a recommender system is also used to provide recommendations on books to read, movies to watch, courses to take or websites to visit. Similarity between users is an important impact for recommendation, which could be calculated from the data of past user ratings of the item by methods of collaborative filtering, matrix factorization or singular vector decomposition. In the development of graph data mining techniques, the relationships between users and items can be represented by matrices from which collaborative filtering could be done with the larger database, more accurate and faster in calculation. All these data can be represented graphically and mined by today's highly developed graph neural network models. On the other hand, users' social friendship data also influence consumption habits because recommendations from friends will be considered more carefully than information sources. However, combining a user's friend influence and the similarity between users whose similar shopping habits is challenging. Because the information is noisy and it affects each particular data set in different ways. In this study, we present the input data processing method to remove outliers which are single reviews or users with little interaction with the items; the next proposed model will combine the social relationship data and the similarity in the rating history of users to improve the accuracy and recall of the recommender system.

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arXiv2025

Improvement Graph Convolution Collaborative Filtering with Weighted addition input

Graph Neural Networks have been extensively applied in the field of machine learning to find features of graphs, and recommendation systems are no exception. The ratings of users on considered items can be represented by graphs which are input for many efficient models to find out the characteristics of the users and the items. From these insights, relevant items are recommended to users. However, user's decisions on the items have varying degrees of effects on different users, and this information should be learned so as not to be lost in the process of information mining. In this publication, we propose to build an additional graph showing the recommended weight of an item to a target user to improve the accuracy of GNN models. Although the users' friendships were not recorded, their correlation was still evident through the commonalities in consumption behavior. We build a model WiGCN (Weighted input GCN) to describe and experiment on well-known datasets. Conclusions will be stated after comparing our results with state-of-the-art such as GCMC, NGCF and LightGCN. The source code is also included at https://github.com/trantin84/WiGCN.

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arXiv2025

CombiGCN: An effective GCN model for Recommender System

Graph Neural Networks (GNNs) have opened up a potential line of research for collaborative filtering (CF). The key power of GNNs is based on injecting collaborative signal into user and item embeddings which will contain information about user-item interactions after that. However, there are still some unsatisfactory points for a CF model that GNNs could have done better. The way in which the collaborative signal are extracted through an implicit feedback matrix that is essentially built on top of the message-passing architecture of GNNs, and it only helps to update the embedding based on the value of the items (or users) embeddings neighboring. By identifying the similarity weight of users through their interaction history, a key concept of CF, we endeavor to build a user-user weighted connection graph based on their similarity weight. In this study, we propose a recommendation framework, CombiGCN, in which item embeddings are only linearly propagated on the user-item interaction graph, while user embeddings are propagated simultaneously on both the user-user weighted connection graph and user-item interaction graph graphs with Light Graph Convolution (LGC) and combined in a simpler method by using the weighted sum of the embeddings for each layer. We also conducted experiments comparing CombiGCN with several state-of-the-art models on three real-world datasets.

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arXiv2025

BeLightRec: A lightweight recommender system enhanced with BERT

The trend of data mining using deep learning models on graph neural networks has proven effective in identifying object features through signal encoders and decoders, particularly in recommendation systems utilizing collaborative filtering methods. Collaborative filtering exploits similarities between users and items from historical data. However, it overlooks distinctive information, such as item names and descriptions. The semantic data of items should be further mined using models in the natural language processing field. Thus, items can be compared using text classification, similarity assessments, or identifying analogous sentence pairs. This research proposes combining two sources of item similarity signals: one from collaborative filtering and one from the semantic similarity measure between item names and descriptions. These signals are integrated into a graph convolutional neural network to optimize model weights, thereby providing accurate recommendations. Experiments are also designed to evaluate the contribution of each signal group to the recommendation results.

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arXiv2022

Remarks on scalable frames

This paper investigates scalable frame in ${\mathbb R}^n$. We define the reduced diagram matrix of a frame and use it to classify scalability of the frame under some conditions. We give a new approach to the scaling problem by breaking the problem into two smaller ones, each of which is easily solved, giving a simple way to check scaling. Finally, we study the scalability of dual frames.

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arXiv2022

Piecewise scalable frames

In this paper we define "piecewise scalable frames". This new scaling process allows us to alter many frames to Parseval frames which is impossible by the previous standard scaling. We give necessary and sufficient conditions for a frame to be piecewise scalable. We show that piecewise scalability is preserved under unitary transformations. Unlike standard scaling, we show that all frames in $\RR^2$ and $\RR^3$ are piecewise scalable. We also show that if the frame vectors are close to each other, then they might not be piecewise scalable. Several properties of scaling constants are also presented.

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arXiv2022

The core of a Grassmannian frame

Let $X=\{x_i\}_{i=1}^m$ be a set of unit vectors in $\RR^n$. The coherence of $X$ is $\coh(X):=\max_{i\not=j}|\langle x_i, x_j\rangle|$. A vector $x\in X$ is said to be isolable if there are no unit vectors $x'$ arbitrarily close to $x$ such that $|\langle x', y\rangle|<\coh(X)$ for all other vectors $y$ in $X$. We define the {\bf core} of a Grassmannian frame $X=\{x_i\}_{i=1}^m$ in $\RR^n$ at angle $α$ as a maximal subset of $X$ which has coherence $α$ and has no isolable vectors. In other words, if $Y$ is a subset of $X$, $\coh(Y)=α$, and $Y$ has no isolable vectors, then $Y$ is a subset of the core. We will show that every Grassmannian frame of $m>n$ vectors for $\RR^n$ has the property that each vector in the core makes angle $α$ with a spanning family from the core. Consequently, the core consists of $\ge n+1$ vectors. We then develop other properties of Grassmannian frames and of the core.

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arXiv2019

A notion of optimal packings of subspaces with mixed-rank and solutions

We resolve a longstanding open problem by reformulating the Grassmannian fusion frames to the case of mixed dimensions and show that this satisfies the proper properties for the problem. In order to compare elements of mixed dimension, we use a classical embedding to send all fusion frame elements to points on a higher dimensional Euclidean sphere, where they are given "equal footing". Over the embedded images -- a compact subset in the higher dimensional embedded sphere -- we define optimality in terms of the corresponding restricted coding problem. We then construct infinite families of solutions to the problem by using maximal sets of mutually unbiased bases and block designs. Finally, we show that using Hadamard 3-designs in this construction leads to infinite examples of maximal orthoplectic fusion frames of constant-rank. Moreover, any such fusion frames constructed by this method must come from Hadamard 3-designs.

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arXiv2019

Regular two distance sets

This paper makes a deep study of regular two-distance sets. A set of unit vectors $X$ in Euclidean space $\RR^n$ is said to be regular two-distance set if the inner product of any pair of its vectors is either $α$ or $β$, and the number of $α$ (and hence $β$) on each row of the Gram matrix of $X$ are the same. We present various properties of these sets as well as focus on the case where they form tight frames for the underling space. We then give some constructions of regular two-distance sets, in particular, two-distance frames, both tight and non-tight cases. Connections among two-distance sets, equiangular lines and quasi-symmetric designs are also given.

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arXiv2018

Constructions and properties of optimally spread subspace packings via symmetric and affine block designs and mutually unbiased bases

We continue the study of optimal chordal packings, with emphasis on packing subspaces of dimension greater than one. Following a principle outlined in a previous work, where the authors use maximal affine block designs and maximal sets of mutually unbiased bases to construct Grassmannian $2$-designs, we show that their method extends to other types of block designs, leading to a plethora of optimal subspace packings characterized by the orthoplex bound. More generally, we show that any optimal chordal packing is necessarily a fusion frame and that its spatial complement is also optimal.

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arXiv2018

The Solution to the Frame Quantum Detection Problem

We will give a complete solution to the frame quantum detection problem. We will solve both cases of the problem: the quantum injectivity problem and quantum state estimation problem. We will answer the problem in both the real and complex cases and in both the finite dimensional and infinite dimensional cases. Finite Dimensional Case: (1) We give two complete classifications of the sets of vectors which solve the injectivity problem - for both the real and complex cases. We also give methods for constructing them. (2) We show that the frames which solve the injectivity problem are open and dense in the family of all frames. (3) We show that the Parseval frames which give injectivity are dense in the Parseval frames. (4) We classify all frames for which the state estimation problem is solvable, and when it is not solvable, we give the best approximation to a solution. Infinite Dimensional Case: (1) We give a classification of all frames which solve the injectivity problem and give methods for constructing solutions. (2) We show that the frames solving the injectivity problem are neither open nor dense in all frames. (3) We give necessary and sufficient conditions for the state estimation problem to always have solutions for measurements in $\ell_1$ and show that there is no injective frame for which the state estimation problem is solvable for all measurements in $\ell_2$. (4) When the state estimation problem does not have an exact solution, we give the best approximation to a solution.

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arXiv2018

Phase Retrieval in $\ell_2(\RR)$

We will review the major results in finite dimensional real phase retrieval for vectors and projections. We then (1)prove that many of these theorems hold in infinite dimensions, (2) give counter-examples to show that many others fail in infinite dimensions, (3)list finite dimensional results are unknown for $\ell_2$.

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arXiv2017

Toward the Classification of Biangular Harmonic Frames

Equiangular tight frames (ETFs) and biangular tight frames (BTFs) - sets of unit vectors with basis-like properties whose pairwise absolute inner products admit exactly one or two values, respectively - are useful for many applications. A well-understood class of ETFs are those which manifest as harmonic frames - vector sets defined in terms of the characters of finite abelian groups - because they are characterized by combinatorial objects called difference sets. This work is dedicated to the study of the underlying combinatorial structures of harmonic BTFs. We show that if a harmonic frame is generated by a divisible difference set, a partial difference set or by a special structure with certain Gauss summing properties - all three of which are generalizations of difference sets that fall under the umbrella term "bidifference set" - then it is either a BTF or an ETF. However, we also show that the relationship between harmonic BTFs and bidifference sets is not as straightforward as the correspondence between harmonic ETFs and difference sets, as there are examples of bidifference sets that do not generate harmonic BTFs. In addition, we study another class of combinatorial structures, the nested divisible difference sets, which yields an example of a harmonic BTF that is not generated by a bidifference set.

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arXiv2017

Phase retrieval by hyperplanes

We show that a scalable frame does phase retrieval if and only if the hyperplanes of its orthogonal complements do phase retrieval. We then show this result fails in general by giving an example of a frame for $\mathbb R^3$ which does phase retrieval but its induced hyperplanes fail phase retrieval. Moreover, we show that such frames always exist in $\mathbb R^d$ for any dimension $d$. We also give an example of a frame in $\mathbb R^3$ which fails phase retrieval but its perps do phase retrieval. We will also see that a family of hyperplanes doing phase retrieval in $\mathbb R^d$ must contain at least $2d-2$ hyperplanes. Finally, we provide an example of six hyperplanes in $\mathbb R^4$ which do phase retrieval.

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