Efficient Generative Modeling with Unitary Matrix Product States Using Riemannian Optimization

Haotong Duan; Zhongming Chen; Ngai Wong

cs.LGMar 12, 2026

Efficient Generative Modeling with Unitary Matrix Product States Using Riemannian Optimization

Authors: Haotong Duan, Zhongming Chen, Ngai Wong

Organizations: Department of Mathematics, School of Sciences, Hangzhou Dianzi University, Hangzhou 310018, China. · Department of Electrical and Electronic Engineering, The University of Hong Kong, Hong Kong.

Abstract

Tensor networks, which are originally developed for characterizing complex quantum many-body systems, have recently emerged as a powerful framework for capturing high-dimensional probability distributions with strong physical interpretability. This paper systematically studies matrix product states (MPS) for generative modeling and shows that unitary MPS, which is a tensor-network architecture that is both simple and expressive, offers clear benefits for unsupervised learning by reducing ambiguity in parameter updates and improving efficiency. To overcome the inefficiency of standard gradient-based MPS training, we develop a Riemannian optimization approach that casts probabilistic modeling as an optimization problem with manifold constraints, and further derive an efficient space-decoupling algorithm. Experiments on Bars-and-Stripes and EMNIST datasets demonstrate fast adaptation to data structure, stable updates, and strong performance while maintaining the efficiency and expressive power of MPS.

Figures & tables

Panel of Figure 1 — Fig. 1: Optimization flowchart of the space-decoupling method, where dashed lines indicate the optimization process.

Operation	Time complexity
Two SVDs (lines 5 and 10)	$O(2nr_{\max}^{2}+r_{\max}^{3})$
Compute $\nabla f(X)$ via ( 22 )	$O(\|\mathcal{T}\|dn^{2}r_{\max}^{2})$
Compute $(\bar{K},\bar{V}_{p})$ via ( 26 ) (line 8)	$O(n^{2}r_{\max}^{3})$
Update $X$ (line 9)	$O(n^{2}r_{\max}^{3}))$
Object	Space complexity
MPS cores $A^{(1)},\dots,A^{(d)}$	$dnr_{\max}^{2}$

TABLE I: Summary of time and space complexity for the main operations and objects of Algorithm 1 .

Panel of Figure 3 — Fig. 2: Schematic of sequential sample generation from a UMPS. All tensors except $A^{(d)}$ are gauged to be left-canonical, Each tensor $A^{(i)}$ generates one bit $v_{i}$ conditionally on the bits to its right.

Panel of Figure 4 — Fig. 3: (a) Illustration of column-major flattening of a $16\times 16$ image into a 256-dimensional vector $v=(p_{1},p_{2},\dots,p_{16},p_{17},\dots,p_{256})^{\top}$ . (b) A subset of the Bars-and-Stripes dataset, with images of size $16\times 16$ .

Panel of Figure 5 — Fig. 4: (a) The blue curve represents the average NLL as a function of the number of loops, while the orange bars indicate the cumulative computation time required to reach the corresponding loops. (b) Images generated by the model at loops from 1 to 5 as indicated in (a). In experiments above, we set $r_{\rm max}=500$ , $|\mathcal{T}|=400$ , and the learning rate to $0.007$ .

$loops$	1	2	3	4	5
$r_{\rm mean}$	7.8945	31.1133	122.4883	470.7852	470.7852
$r_{\rm max}$	8	32	128	500	500

TABLE II: Under the experimental conditions shown in Fig. 4 , as the number of loops increases, the table shows the average bond dimension

r_{\rm mean}

and the largest bond dimension

r_{\rm max}

in the UMPS updated up to this step.

Panel of Figure 7 — Fig. 5: (a) Average NLL as a function of the dataset size under different $r_{\rm max}$ . For each curve, the circular markers from left to right correspond to the NLL values obtained at $|\mathcal{T}|=100,200,300,400,$ and $500$ , respectively. (b) The average computation time per experiment shown in (a).

Panel of Figure 8 — Fig. 6: Panels (a) and (b) correspond to the negative log-likelihood (NLL) convergence of the optimization algorithm of the MPS model and UMPS-SD algorithm for maximum bond dimensions $r_{\rm max}=200$ and $r_{\rm max}=400$ , respectively, with a training set of $|\mathcal{T}|=100$ samples. The blue line represents the optimization algorithm of the MPS model, and the red line represents the UMPS-SD algorithm.

$loops$	MPS		UMPS-SD
	NLL	Time/s	NLL	Time/s
1	154.74	1.30	167.70	1.68
2	94.48	3.14	80.69	5.61
3	62.25	7.86	13.01	30.25
4	46.85	26.08	-	-
5	40.14	53.61	-	-

TABLE III: Under the same experimental conditions as Fig. 6a , NLL and training time (s) for MPS and UMPS-SD models with learning rate

1\times 10^{-3}

. The time metric indicates the cumulative time required to reach this NLL value.

$loops$	UMPS-SD(1)		UMPS-SD(2)		UMPS-SD(3)
	NLL	Time/s	NLL	Time/s	NLL	Time/s
1	171.99	1.62	164.41	1.59	165.76	1.62
2	79.90	5.59	81.14	5.48	79.97	5.55
3	7.35	30.73	6.568	30.23	12.30	30.11

TABLE IV: Under the same experimental conditions as Fig. 6a , from (1) to (3), the learning rates are set to

2.5\times 10^{-4}

5\times 10^{-4}

, and

2\times 10^{-3}

, respectively.

Panel of Figure 11 — Fig. 7: (a) Generated from the MPS model with a maximum bond dimension $r_{\max}=400$ , trained on a subset of $|\mathcal{T}|=300$ samples randomly selected from the EMNIST dataset for $l_{\max}=25$ . (b) Generated by the proposed UMPS model with $l_{\max}=4$ , with all other settings kept identical to (a).

Panel of Figure 12 — Fig. 8: Image reconstruction from half of the images, which are selected from the training set. For both experiments, the maximum bond dimension is set to $r_{\rm max}=200$ and $|\mathcal{T}|=100$ . The update details are summarized in Table III . The black areas in the image are the given pixels, and the blue areas are the pixels completed by the model. (a) The image reconstructed by the MPS model; (b) The image reconstructed by the UMPS model.

Fig. 9: The curves illustrate how the NLL varies with the bond dimension $r_{\rm max}$ , where the training set $\mathcal{T}$ is selected from the EMNIST-Letters dataset with a fixed size of $|\mathcal{T}|=150$ . All marked points correspond to the setting of $l_{\max}=4$ . In the images, black pixels denote the given observations, while the blue pixels are reconstructed by the UMPS model. The figure presents the reconstruction results obtained for $r_{\rm max}=120$ , $150$ , and $175$ , respectively.

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