On a class of inverse quadratic eigenvalue problem

AbstractIn this paper, we first give the representation of the general solution of the following inverse monic quadratic eigenvalue problem (IMQEP): given matrices Λ=diag{λ1,…,λp}∈Cp×p, λi≠λj for i≠j, i,j=1,…,p, X=[x1,…,xp]∈Cn×p, rank(X)=p, and both Λ and X are closed under complex conjugation in the sense that λ2j=λ̄2j−1∈C, x2j=x̄2j−1∈Cn for j=1,…,l, and λk∈R, xk∈Rn for k=2l+1,…,p, find real-valued symmetric matrices D and K such that XΛ2+DXΛ+KX=0. Then we consider a best approximation problem: given D̃,K̃∈Rn×n, find (Dˆ,Kˆ)∈SDK such that ‖(Dˆ,Kˆ)−(D̃,K̃)‖W=min(D,K)∈SDK‖(D,K)−(D̃,K̃)‖W, where ‖⋅‖W is a weighted Frobenius norm and SDK is the solution set of IMQEP. We show that the best approximation solution (Dˆ,Kˆ) is unique and derive an explicit formula for it

Solutions to an inverse monic quadratic eigenvalue problem

AbstractGiven n+1 pairs of complex numbers and vectors (closed under complex conjugation), the inverse quadratic eigenvalue problem is to construct real symmetric or anti-symmetric matrix C and real symmetric matrix K of size n×n so that the quadratic pencil Q(λ)=λ2In+λC+K has the given n+1 pairs as eigenpairs. Necessary and sufficient conditions under which this quadratic inverse eigenvalue problem is solvable are obtained. Numerical algorithms for solving the problem are developed. Numerical examples illustrating these solutions are presented

An Inverse Quadratic Eigenvalue Problem for Damped Structural Systems

We first give the representation of the general solution of the following inverse quadratic eigenvalue problem (IQEP): given Λ=diag{λ1,…,λp}∈Cp×p , X=[x1,…,xp]∈Cn×p, and both Λ and X are closed under complex conjugation in the sense that λ2j=λ¯2j−1∈C, x2j=x¯2j−1∈Cn for j=1,…,l, and λk∈R, xk∈Rn for k=2l+1,…, p, find real-valued symmetric (2r+1)-diagonal matrices M, D and K such that MXΛ2+DXΛ+KX=0. We then consider an optimal approximation problem: given real-valued symmetric (2r+1)-diagonal matrices Ma,Da,Ka∈Rn×n, find (M^,D^,K^)∈SE such that ‖M^−Ma‖2+‖D^−Da‖2+‖K^−Ka‖2=infâ¡(M,D,K)∈SE(‖M−Ma‖2+‖D−Da‖2+‖K−Ka‖2), where SE is the solution set of IQEP. We show that the optimal approximation solution (M^,D^,K^) is unique and derive an explicit formula for it

Stiffness matrix modification with vibration test data by displacement feedback technique

A no spill-over method is developed which uses measured normal modes and natural frequencies to adjust a structural dynamics model in light of displacement feedback technique. By the method, the required displacement feedback gain matrix is determined, and thus the updated stiffness matrix which satisfies the characteristic equation is found in the Frobenius norm sense and the large number of unmeasured high-order modal data of the original model is preserved. The method directly identifies, without iteration, and the solution of this problem is of a compact expression. The numerical example shows that the modal measured data are better incorporated into the updated model

Nesterov smoothing for sampling without smoothness

We study the problem of sampling from a target distribution in $\mathbb{R}^d$ whose potential is not smooth. Compared with the sampling problem with smooth potentials, this problem is much less well-understood due to the lack of smoothness. In this paper, we propose a novel sampling algorithm for a class of non-smooth potentials by first approximating them by smooth potentials using a technique that is akin to Nesterov smoothing. We then utilize sampling algorithms on the smooth potentials to generate approximate samples from the original non-smooth potentials. We select an appropriate smoothing intensity to ensure that the distance between the smoothed and un-smoothed distributions is minimal, thereby guaranteeing the algorithm's accuracy. Hence we obtain non-asymptotic convergence results based on existing analysis of smooth sampling. We verify our convergence result on a synthetic example and apply our method to improve the worst-case performance of Bayesian inference on a real-world example

Where to Go Next for Recommender Systems? ID- vs. Modality-based Recommender Models Revisited

Recommendation models that utilize unique identities (IDs) to represent distinct users and items have been state-of-the-art (SOTA) and dominated the recommender systems (RS) literature for over a decade. Meanwhile, the pre-trained modality encoders, such as BERT and ViT, have become increasingly powerful in modeling the raw modality features of an item, such as text and images. Given this, a natural question arises: can a purely modality-based recommendation model (MoRec) outperforms or matches a pure ID-based model (IDRec) by replacing the itemID embedding with a SOTA modality encoder? In fact, this question was answered ten years ago when IDRec beats MoRec by a strong margin in both recommendation accuracy and efficiency. We aim to revisit this `old' question and systematically study MoRec from several aspects. Specifically, we study several sub-questions: (i) which recommendation paradigm, MoRec or IDRec, performs better in practical scenarios, especially in the general setting and warm item scenarios where IDRec has a strong advantage? does this hold for items with different modality features? (ii) can the latest technical advances from other communities (i.e., natural language processing and computer vision) translate into accuracy improvement for MoRec? (iii) how to effectively utilize item modality representation, can we use it directly or do we have to adjust it with new data? (iv) are there some key challenges for MoRec to be solved in practical applications? To answer them, we conduct rigorous experiments for item recommendations with two popular modalities, i.e., text and vision. We provide the first empirical evidence that MoRec is already comparable to its IDRec counterpart with an expensive end-to-end training method, even for warm item recommendation. Our results potentially imply that the dominance of IDRec in the RS field may be greatly challenged in the future

FMMRec: Fairness-aware Multimodal Recommendation

Recently, multimodal recommendations have gained increasing attention for effectively addressing the data sparsity problem by incorporating modality-based representations. Although multimodal recommendations excel in accuracy, the introduction of different modalities (e.g., images, text, and audio) may expose more users' sensitive information (e.g., gender and age) to recommender systems, resulting in potentially more serious unfairness issues. Despite many efforts on fairness, existing fairness-aware methods are either incompatible with multimodal scenarios, or lead to suboptimal fairness performance due to neglecting sensitive information of multimodal content. To achieve counterfactual fairness in multimodal recommendations, we propose a novel fairness-aware multimodal recommendation approach (dubbed as FMMRec) to disentangle the sensitive and non-sensitive information from modal representations and leverage the disentangled modal representations to guide fairer representation learning. Specifically, we first disentangle biased and filtered modal representations by maximizing and minimizing their sensitive attribute prediction ability respectively. With the disentangled modal representations, we mine the modality-based unfair and fair (corresponding to biased and filtered) user-user structures for enhancing explicit user representation with the biased and filtered neighbors from the corresponding structures, followed by adversarially filtering out sensitive information. Experiments on two real-world public datasets demonstrate the superiority of our FMMRec relative to the state-of-the-art baselines. Our source code is available at https://anonymous.4open.science/r/FMMRec

HeteFedRec: Federated Recommender Systems with Model Heterogeneity

Owing to the nature of privacy protection, federated recommender systems (FedRecs) have garnered increasing interest in the realm of on-device recommender systems. However, most existing FedRecs only allow participating clients to collaboratively train a recommendation model of the same public parameter size. Training a model of the same size for all clients can lead to suboptimal performance since clients possess varying resources. For example, clients with limited training data may prefer to train a smaller recommendation model to avoid excessive data consumption, while clients with sufficient data would benefit from a larger model to achieve higher recommendation accuracy. To address the above challenge, this paper introduces HeteFedRec, a novel FedRec framework that enables the assignment of personalized model sizes to participants. In HeteFedRec, we present a heterogeneous recommendation model aggregation strategy, including a unified dual-task learning mechanism and a dimensional decorrelation regularization, to allow knowledge aggregation among recommender models of different sizes. Additionally, a relation-based ensemble knowledge distillation method is proposed to effectively distil knowledge from heterogeneous item embeddings. Extensive experiments conducted on three real-world recommendation datasets demonstrate the effectiveness and efficiency of HeteFedRec in training federated recommender systems under heterogeneous settings