NASep 7, 2011
Trace Norm Regularized Tensor Classification and Its Online Learning ApproachesZiqiang Shi, Tieran Zheng, Jiqing Han
In this paper we propose an algorithm to classify tensor data. Our methodology is built on recent studies about matrix classification with the trace norm constrained weight matrix and the tensor trace norm. Similar to matrix classification, the tensor classification is formulated as a convex optimization problem which can be solved by using the off-the-shelf accelerated proximal gradient (APG) method. However, there are no analytic solutions as the matrix case for the updating of the weight tensors via the proximal gradient. To tackle this problem, the Douglas-Rachford splitting technique and the alternating direction method of multipliers (ADM) used in tensor completion are adapted to update the weight tensors. Further more, due to the demand of real applications, we also propose its online learning approaches. Experiments demonstrate the efficiency of the methods.
NADec 21, 2016
Online and stochastic Douglas-Rachford splitting method for large scale machine learningZiqiang Shi, Rujie Liu
Online and stochastic learning has emerged as powerful tool in large scale optimization. In this work, we generalize the Douglas-Rachford splitting (DRs) method for minimizing composite functions to online and stochastic settings (to our best knowledge this is the first time DRs been generalized to sequential version). We first establish an $O(1/\sqrt{T})$ regret bound for batch DRs method. Then we proved that the online DRs splitting method enjoy an $O(1)$ regret bound and stochastic DRs splitting has a convergence rate of $O(1/\sqrt{T})$. The proof is simple and intuitive, and the results and technique can be served as a initiate for the research on the large scale machine learning employ the DRs method. Numerical experiments of the proposed method demonstrate the effectiveness of the online and stochastic update rule, and further confirm our regret and convergence analysis.
CVFeb 10
Scalpel: Fine-Grained Alignment of Attention Activation Manifolds via Mixture Gaussian Bridges to Mitigate Multimodal HallucinationZiqiang Shi, Rujie Liu, Shanshan Yu et al.
Rapid progress in large vision-language models (LVLMs) has achieved unprecedented performance in vision-language tasks. However, due to the strong prior of large language models (LLMs) and misaligned attention across modalities, LVLMs often generate outputs inconsistent with visual content - termed hallucination. To address this, we propose \textbf{Scalpel}, a method that reduces hallucination by refining attention activation distributions toward more credible regions. Scalpel predicts trusted attention directions for each head in Transformer layers during inference and adjusts activations accordingly. It employs a Gaussian mixture model to capture multi-peak distributions of attention in trust and hallucination manifolds, and uses entropic optimal transport (equivalent to Schrödinger bridge problem) to map Gaussian components precisely. During mitigation, Scalpel dynamically adjusts intervention strength and direction based on component membership and mapping relationships between hallucination and trust activations. Extensive experiments across multiple datasets and benchmarks demonstrate that Scalpel effectively mitigates hallucinations, outperforming previous methods and achieving state-of-the-art performance. Moreover, Scalpel is model- and data-agnostic, requiring no additional computation, only a single decoding step.
CVFeb 10
SchröMind: Mitigating Hallucinations in Multimodal Large Language Models via Solving the Schrödinger Bridge ProblemZiqiang Shi, Rujie Liu, Shanshan Yu et al.
Recent advancements in Multimodal Large Language Models (MLLMs) have achieved significant success across various domains. However, their use in high-stakes fields like healthcare remains limited due to persistent hallucinations, where generated text contradicts or ignores visual input. We contend that MLLMs can comprehend images but struggle to produce accurate token sequences. Minor perturbations can shift attention from truthful to untruthful states, and the autoregressive nature of text generation often prevents error correction. To address this, we propose SchröMind-a novel framework reducing hallucinations via solving the Schrödinger bridge problem. It establishes a token-level mapping between hallucinatory and truthful activations with minimal transport cost through lightweight training, while preserving the model's original capabilities. Extensive experiments on the POPE and MME benchmarks demonstrate the superiority of Schrödinger, which achieves state-of-the-art performance while introducing only minimal computational overhead.
ASSep 7, 2020Code
Toward Speech Separation in The Pre-Cocktail Party Problem with TasTasZiqiang Shi, Jiqing Han
In this note, we propose to use TasTas \cite{shi2020speech} for the end-to-end approach to monaural speech separation in the pre-cocktail party problem. Our experiments on the public WSJ0-5mix data corpus results in 10.41dB SDR improvement. If online voice data remixing augmentation \cite{zeghidour2020wavesplit} is adopted in training, an 11.14dB SDR improvement can be achieved. We have open-sourced our re-implementation of the DPRNN-TasNet in https://github.com/ShiZiqiang/dual-path-RNNs-DPRNNs-based-speech-separation, and our TasTas is realized based on this implementation of DPRNN-TasNet, it is believed that the results in this paper can be reproduced with ease.
ASAug 6, 2020Code
Speech Separation Based on Multi-Stage Elaborated Dual-Path Deep BiLSTM with Auxiliary Identity LossZiqiang Shi, Rujie Liu, Jiqing Han
Deep neural network with dual-path bi-directional long short-term memory (BiLSTM) block has been proved to be very effective in sequence modeling, especially in speech separation. This work investigates how to extend dual-path BiLSTM to result in a new state-of-the-art approach, called TasTas, for multi-talker monaural speech separation (a.k.a cocktail party problem). TasTas introduces two simple but effective improvements, one is an iterative multi-stage refinement scheme, and the other is to correct the speech with imperfect separation through a loss of speaker identity consistency between the separated speech and original speech, to boost the performance of dual-path BiLSTM based networks. TasTas takes the mixed utterance of two speakers and maps it to two separated utterances, where each utterance contains only one speaker's voice. Our experiments on the notable benchmark WSJ0-2mix data corpus result in 20.55dB SDR improvement, 20.35dB SI-SDR improvement, 3.69 of PESQ, and 94.86\% of ESTOI, which shows that our proposed networks can lead to big performance improvement on the speaker separation task. We have open sourced our re-implementation of the DPRNN-TasNet here (https://github.com/ShiZiqiang/dual-path-RNNs-DPRNNs-based-speech-separation), and our TasTas is realized based on this implementation of DPRNN-TasNet, it is believed that the results in this paper can be reproduced with ease.
SDJan 23, 2020Code
LaFurca: Iterative Refined Speech Separation Based on Context-Aware Dual-Path Parallel Bi-LSTMZiqiang Shi, Rujie Liu, Jiqing Han
Deep neural network with dual-path bi-directional long short-term memory (BiLSTM) block has been proved to be very effective in sequence modeling, especially in speech separation, e.g. DPRNN-TasNet \cite{luo2019dual}. In this paper, we propose several improvements of dual-path BiLSTM based network for end-to-end approach to monaural speech separation. Firstly a dual-path network with intra-parallel BiLSTM and inter-parallel BiLSTM components is introduced to reduce performance sub-variances among different branches. Secondly, we propose to use global context aware inter-intra cross-parallel BiLSTM to further perceive the global contextual information. Finally, a spiral multi-stage dual-path BiLSTM is proposed to iteratively refine the separation results of the previous stages. All these networks take the mixed utterance of two speakers and map it to two separate utterances, where each utterance contains only one speaker's voice. For the objective, we propose to train the network by directly optimizing the utterance level scale-invariant signal-to-distortion ratio (SI-SDR) in a permutation invariant training (PIT) style. Our experiments on the public WSJ0-2mix data corpus results in 20.55dB SDR improvement, 20.35dB SI-SDR improvement, 3.69 of PESQ, and 94.86\% of ESTOI, which shows our proposed networks can lead to performance improvement on the speaker separation task. We have open-sourced our re-implementation of the DPRNN-TasNet in https://github.com/ShiZiqiang/dual-path-RNNs-DPRNNs-based-speech-separation, and our LaFurca is realized based on this implementation of DPRNN-TasNet, it is believed that the results in this paper can be reproduced with ease.
LGApr 19, 2024
Generative Modelling with High-Order Langevin DynamicsZiqiang Shi, Rujie Liu
Diffusion generative modelling (DGM) based on stochastic differential equations (SDEs) with score matching has achieved unprecedented results in data generation. In this paper, we propose a novel fast high-quality generative modelling method based on high-order Langevin dynamics (HOLD) with score matching. This motive is proved by third-order Langevin dynamics. By augmenting the previous SDEs, e.g. variance exploding or variance preserving SDEs for single-data variable processes, HOLD can simultaneously model position, velocity, and acceleration, thereby improving the quality and speed of the data generation at the same time. HOLD is composed of one Ornstein-Uhlenbeck process and two Hamiltonians, which reduce the mixing time by two orders of magnitude. Empirical experiments for unconditional image generation on the public data set CIFAR-10 and CelebA-HQ show that the effect is significant in both Frechet inception distance (FID) and negative log-likelihood, and achieves the state-of-the-art FID of 1.85 on CIFAR-10.
SDJan 29, 2022
ItôWave: Itô Stochastic Differential Equation Is All You Need For Wave GenerationShoule Wu, Ziqiang Shi
In this paper, we propose a vocoder based on a pair of forward and reverse-time linear stochastic differential equations (SDE). The solutions of this SDE pair are two stochastic processes, one of which turns the distribution of wave, that we want to generate, into a simple and tractable distribution. The other is the generation procedure that turns this tractable simple signal into the target wave. The model is called ItôWave. ItôWave use the Wiener process as a driver to gradually subtract the excess signal from the noise signal to generate realistic corresponding meaningful audio respectively, under the conditional inputs of original mel spectrogram. The results of the experiment show that the mean opinion scores (MOS) of ItôWave can exceed the current state-of-the-art (SOTA) methods, and reached 4.35$\pm$0.115. The generated audio samples are available online.
CVJul 7, 2021
Multi-modal Affect Analysis using standardized data within subjects in the WildSachihiro Youoku, Takahisa Yamamoto, Junya Saito et al.
Human affective recognition is an important factor in human-computer interaction. However, the method development with in-the-wild data is not yet accurate enough for practical usage. In this paper, we introduce the affective recognition method focusing on facial expression (EXP) and valence-arousal calculation that was submitted to the Affective Behavior Analysis in-the-wild (ABAW) 2021 Contest. When annotating facial expressions from a video, we thought that it would be judged not only from the features common to all people, but also from the relative changes in the time series of individuals. Therefore, after learning the common features for each frame, we constructed a facial expression estimation model and valence-arousal model using time-series data after combining the common features and the standardized features for each video. Furthermore, the above features were learned using multi-modal data such as image features, AU, Head pose, and Gaze. In the validation set, our model achieved a facial expression score of 0.546. These verification results reveal that our proposed framework can improve estimation accuracy and robustness effectively.
SDMay 17, 2021
ItôTTS and ItôWave: Linear Stochastic Differential Equation Is All You Need For Audio GenerationShoule Wu, Ziqiang Shi
In this paper, we propose to unify the two aspects of voice synthesis, namely text-to-speech (TTS) and vocoder, into one framework based on a pair of forward and reverse-time linear stochastic differential equations (SDE). The solutions of this SDE pair are two stochastic processes, one of which turns the distribution of mel spectrogram (or wave), that we want to generate, into a simple and tractable distribution. The other is the generation procedure that turns this tractable simple signal into the target mel spectrogram (or wave). The model that generates mel spectrogram is called ItôTTS, and the model that generates wave is called ItôWave. ItôTTS and ItôWave use the Wiener process as a driver to gradually subtract the excess signal from the noise signal to generate realistic corresponding meaningful mel spectrogram and audio respectively, under the conditional inputs of original text or mel spectrogram. The results of the experiment show that the mean opinion scores (MOS) of ItôTTS and ItôWave can exceed the current state-of-the-art methods, and reached 3.925$\pm$0.160 and 4.35$\pm$0.115 respectively. The generated audio samples are available at https://wushoule.github.io/ItoAudio/. All authors contribute equally to this work.
CVSep 23, 2020
HiCOMEX: Facial Action Unit Recognition Based on Hierarchy Intensity Distribution and COMEX Relation LearningZiqiang Shi, Liu Liu, Zhongling Liu et al.
The detection of facial action units (AUs) has been studied as it has the competition due to the wide-ranging applications thereof. In this paper, we propose a novel framework for the AU detection from a single input image by grasping the \textbf{c}o-\textbf{o}ccurrence and \textbf{m}utual \textbf{ex}clusion (COMEX) as well as the intensity distribution among AUs. Our algorithm uses facial landmarks to detect the features of local AUs. The features are input to a bidirectional long short-term memory (BiLSTM) layer for learning the intensity distribution. Afterwards, the new AU feature continuously passed through a self-attention encoding layer and a continuous-state modern Hopfield layer for learning the COMEX relationships. Our experiments on the challenging BP4D and DISFA benchmarks without any external data or pre-trained models yield F1-scores of 63.7\% and 61.8\% respectively, which shows our proposed networks can lead to performance improvement in the AU detection task.
SDFeb 13, 2020
Hodge and Podge: Hybrid Supervised Sound Event Detection with Multi-Hot MixMatch and Composition Consistence TrainingZiqiang Shi, Liu Liu, Huibin Lin et al.
In this paper, we propose a method called Hodge and Podge for sound event detection. We demonstrate Hodge and Podge on the dataset of Detection and Classification of Acoustic Scenes and Events (DCASE) 2019 Challenge Task 4. This task aims to predict the presence or absence and the onset and offset times of sound events in home environments. Sound event detection is challenging due to the lack of large scale real strongly labeled data. Recently deep semi-supervised learning (SSL) has proven to be effective in modeling with weakly labeled and unlabeled data. This work explores how to extend deep SSL to result in a new, state-of-the-art sound event detection method called Hodge and Podge. With convolutional recurrent neural networks (CRNN) as the backbone network, first, a multi-scale squeeze-excitation mechanism is introduced and added to generate a pyramid squeeze-excitation CRNN. The pyramid squeeze-excitation layer can pay attention to the issue that different sound events have different durations, and to adaptively recalibrate channel-wise spectrogram responses. Further, in order to remedy the lack of real strongly labeled data problem, we propose multi-hot MixMatch and composition consistency training with temporal-frequency augmentation. Our experiments with the public DCASE2019 challenge task 4 validation data resulted in an event-based F-score of 43.4\%, and is about absolutely 1.6\% better than state-of-the-art methods in the challenge. While the F-score of the official baseline is 25.8\%.
SDJul 17, 2019
HODGEPODGE: Sound event detection based on ensemble of semi-supervised learning methodsZiqiang Shi, Liu Liu, Huibin Lin et al.
In this paper, we present a method called HODGEPODGE\footnotemark[1] for large-scale detection of sound events using weakly labeled, synthetic, and unlabeled data proposed in the Detection and Classification of Acoustic Scenes and Events (DCASE) 2019 challenge Task 4: Sound event detection in domestic environments. To perform this task, we adopted the convolutional recurrent neural networks (CRNN) as our backbone network. In order to deal with a small amount of tagged data and a large amounts of unlabeled in-domain data, we aim to focus primarily on how to apply semi-supervise learning methods efficiently to make full use of limited data. Three semi-supervised learning principles have been used in our system, including: 1) Consistency regularization applies data augmentation; 2) MixUp regularizer requiring that the predictions for a interpolation of two inputs is close to the interpolation of the prediction for each individual input; 3) MixUp regularization applies to interpolation between data augmentations. We also tried an ensemble of various models, which are trained by using different semi-supervised learning principles. Our proposed approach significantly improved the performance of the baseline, achieving the event-based f-measure of 42.0\% compared to 25.8\% event-based f-measure of the baseline in the provided official evaluation dataset. Our submissions ranked third among 18 teams in the task 4.
CVMar 25, 2019
Learning from Adversarial Features for Few-Shot ClassificationWei Shen, Ziqiang Shi, Jun Sun
Many recent few-shot learning methods concentrate on designing novel model architectures. In this paper, we instead show that with a simple backbone convolutional network we can even surpass state-of-the-art classification accuracy. The essential part that contributes to this superior performance is an adversarial feature learning strategy that improves the generalization capability of our model. In this work, adversarial features are those features that can cause the classifier uncertain about its prediction. In order to generate adversarial features, we firstly locate adversarial regions based on the derivative of the entropy with respect to an averaging mask. Then we use the adversarial region attention to aggregate the feature maps to obtain the adversarial features. In this way, we can explore and exploit the entire spatial area of the feature maps to mine more diverse discriminative knowledge. We perform extensive model evaluations and analyses on miniImageNet and tieredImageNet datasets demonstrating the effectiveness of the proposed method.
SDFeb 12, 2019
FurcaNeXt: End-to-end monaural speech separation with dynamic gated dilated temporal convolutional networksLiwen Zhang, Ziqiang Shi, Jiqing Han et al.
Deep dilated temporal convolutional networks (TCN) have been proved to be very effective in sequence modeling. In this paper we propose several improvements of TCN for end-to-end approach to monaural speech separation, which consists of 1) multi-scale dynamic weighted gated dilated convolutional pyramids network (FurcaPy), 2) gated TCN with intra-parallel convolutional components (FurcaPa), 3) weight-shared multi-scale gated TCN (FurcaSh), 4) dilated TCN with gated difference-convolutional component (FurcaSu), that all these networks take the mixed utterance of two speakers and maps it to two separated utterances, where each utterance contains only one speaker's voice. For the objective, we propose to train the network by directly optimizing utterance level signal-to-distortion ratio (SDR) in a permutation invariant training (PIT) style. Our experiments on the the public WSJ0-2mix data corpus results in 18.4dB SDR improvement, which shows our proposed networks can leads to performance improvement on the speaker separation task.
SDFeb 2, 2019
FurcaNet: An end-to-end deep gated convolutional, long short-term memory, deep neural networks for single channel speech separationZiqiang Shi, Huibin Lin, Liu Liu et al.
Deep gated convolutional networks have been proved to be very effective in single channel speech separation. However current state-of-the-art framework often considers training the gated convolutional networks in time-frequency (TF) domain. Such an approach will result in limited perceptual score, such as signal-to-distortion ratio (SDR) upper bound of separated utterances and also fail to exploit an end-to-end framework. In this paper we present an integrated simple and effective end-to-end approach to monaural speech separation, which consists of deep gated convolutional neural networks (GCNN) that takes the mixed utterance of two speakers and maps it to two separated utterances, where each utterance contains only one speaker's voice. In addition long short-term memory (LSTM) is employed for long term temporal modeling. For the objective, we propose to train the network by directly optimizing utterance level SDR in a permutation invariant training (PIT) style. Our experiments on the public WSJ0-2mix data corpus demonstrate that this new scheme can produce more discriminative separated utterances and leading to performance improvement on the speaker separation task.
SDFeb 2, 2019
Is CQT more suitable for monaural speech separation than STFT? an empirical studyZiqiang Shi, Huibin Lin, Liu Liu et al.
Short-time Fourier transform (STFT) is used as the front end of many popular successful monaural speech separation methods, such as deep clustering (DPCL), permutation invariant training (PIT) and their various variants. Since the frequency component of STFT is linear, while the frequency distribution of human auditory system is nonlinear. In this work we propose and give an empirical study to use an alternative front end called constant Q transform (CQT) instead of STFT to achieve a better simulation of the frequency resolving power of the human auditory system. The upper bound in signal-to-distortion (SDR) of ideal speech separation based on CQT's ideal ration mask (IRM) is higher than that based on STFT. In the same experimental setting on WSJ0-2mix corpus, we examined the performance of CQT under different backends, including the original DPCL, utterance level PIT, and some of their variants. It is found that all CQT-based methods are better than STFT-based methods, and achieved on average 0.4dB better performance than STFT based method in SDR improvements.
SDNov 17, 2017
A Double Joint Bayesian Approach for J-Vector Based Text-dependent Speaker VerificationZiqiang Shi, Mengjiao Wang, Liu Liu et al.
J-vector has been proved to be very effective in text-dependent speaker verification with short-duration speech. However, the current state-of-the-art back-end classifiers, e.g. joint Bayesian model, cannot make full use of such deep features. In this paper, we generalize the standard joint Bayesian approach to model the multi-faceted information in the j-vector explicitly and jointly. In our generalization, the j-vector was modeled as a result derived by a generative Double Joint Bayesian (DoJoBa) model, which contains several kinds of latent variables. With DoJoBa, we are able to explicitly build a model that can combine multiple heterogeneous information from the j-vectors. In verification step, we calculated the likelihood to describe whether the two j-vectors having consistent labels or not. On the public RSR2015 data corpus, the experimental results showed that our approach can achieve 0.02\% EER and 0.02\% EER for impostor wrong and impostor correct cases respectively.
LGApr 20, 2017
Multi-view (Joint) Probability Linear Discrimination Analysis for Multi-view Feature VerificationZiqiang Shi, Liu Liu, Mengjiao Wang et al.
Multi-view feature has been proved to be very effective in many multimedia applications. However, the current back-end classifiers cannot make full use of such features. In this paper, we propose a method to model the multi-faceted information in the multi-view features explicitly and jointly. In our approach, the feature was modeled as a result derived by a generative multi-view (joint\footnotemark[1]) Probability Linear Discriminant Analysis (PLDA) model, which contains multiple kinds of latent variables. The usual PLDA model only considers one single label. However, in practical use, when using multi-task learned network as feature extractor, the extracted feature are always attached to several labels. This type of feature is called multi-view feature. With multi-view (joint) PLDA, we are able to explicitly build a model that can combine multiple heterogeneous information from the multi-view features. In verification step, we calculated the likelihood to describe whether the two features having consistent labels or not. This likelihood are used in the following decision-making. Experiments have been conducted on large scale verification task. On the public RSR2015 data corpus, the results showed that our approach can achieve 0.02\% EER and 0.09\% EER for impostor wrong and impostor correct cases respectively.
OCAug 17, 2016
A better convergence analysis of the block coordinate descent method for large scale machine learningZiqiang Shi, Rujie Liu
This paper considers the problems of unconstrained minimization of large scale smooth convex functions having block-coordinate-wise Lipschitz continuous gradients. The block coordinate descent (BCD) method are among the first optimization schemes suggested for solving such problems \cite{nesterov2012efficiency}. We obtain a new lower (to our best knowledge the lowest currently) bound that is $16p^3$ times smaller than the best known on the information-based complexity of BCD method based on an effective technique called Performance Estimation Problem (PEP) proposed by Drori and Teboulle \cite{drori2012performance} recently for analyzing the performance of first-order black box optimization methods. Numerical test confirms our analysis.
LGApr 18, 2016
Empirical study of PROXTONE and PROXTONE$^+$ for Fast Learning of Large Scale Sparse ModelsZiqiang Shi, Rujie Liu
PROXTONE is a novel and fast method for optimization of large scale non-smooth convex problem \cite{shi2015large}. In this work, we try to use PROXTONE method in solving large scale \emph{non-smooth non-convex} problems, for example training of sparse deep neural network (sparse DNN) or sparse convolutional neural network (sparse CNN) for embedded or mobile device. PROXTONE converges much faster than first order methods, while first order method is easy in deriving and controlling the sparseness of the solutions. Thus in some applications, in order to train sparse models fast, we propose to combine the merits of both methods, that is we use PROXTONE in the first several epochs to reach the neighborhood of an optimal solution, and then use the first order method to explore the possibility of sparsity in the following training. We call such method PROXTONE plus (PROXTONE$^+$). Both PROXTONE and PROXTONE$^+$ are tested in our experiments, and which demonstrate both methods improved convergence speed twice as fast at least on diverse sparse model learning problems, and at the same time reduce the size to 0.5\% for DNN models. The source of all the algorithms is available upon request.
NAAug 22, 2013
Online and stochastic Douglas-Rachford splitting method for large scale machine learningZiqiang Shi, Rujie Liu
Online and stochastic learning has emerged as powerful tool in large scale optimization. In this work, we generalize the Douglas-Rachford splitting (DRs) method for minimizing composite functions to online and stochastic settings (to our best knowledge this is the first time DRs been generalized to sequential version). We first establish an $O(1/\sqrt{T})$ regret bound for batch DRs method. Then we proved that the online DRs splitting method enjoy an $O(1)$ regret bound and stochastic DRs splitting has a convergence rate of $O(1/\sqrt{T})$. The proof is simple and intuitive, and the results and technique can be served as a initiate for the research on the large scale machine learning employ the DRs method. Numerical experiments of the proposed method demonstrate the effectiveness of the online and stochastic update rule, and further confirm our regret and convergence analysis.
ITJul 23, 2012
Guarantees of Augmented Trace Norm Models in Tensor RecoveryZiqiang Shi, Jiqing Han, Tieran Zheng et al.
This paper studies the recovery guarantees of the models of minimizing $\|\mathcal{X}\|_*+\frac{1}{2α}\|\mathcal{X}\|_F^2$ where $\mathcal{X}$ is a tensor and $\|\mathcal{X}\|_*$ and $\|\mathcal{X}\|_F$ are the trace and Frobenius norm of respectively. We show that they can efficiently recover low-rank tensors. In particular, they enjoy exact guarantees similar to those known for minimizing $\|\mathcal{X}\|_*$ under the conditions on the sensing operator such as its null-space property, restricted isometry property, or spherical section property. To recover a low-rank tensor $\mathcal{X}^0$, minimizing $\|\mathcal{X}\|_*+\frac{1}{2α}\|\mathcal{X}\|_F^2$ returns the same solution as minimizing $\|\mathcal{X}\|_*$ almost whenever $α\geq10\mathop {\max}\limits_{i}\|X^0_{(i)}\|_2$.