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Optimization

Mehdi Makni, Xiang Meng, Rahul Mazumder

Sparse plus Low-Rank $(\mathbf{S} + \mathbf{L}\mathbf{R})$ decomposition of Large Language Models (LLMs) has emerged as a promising direction in $\textit{model compression}$, aiming to decompose pre-trained model weights into a sum of sparse and low-rank matrices $\mathbf{W} \approx \mathbf{S} + \mathbf{LR}$. Despite recent progress, existing methods often suffer from substantial performance degradation compared to dense models. In this work, we introduce $\texttt{3BASiL-TM}$, an efficient one-shot post-training method for $(\mathbf{S} + \mathbf{L}\mathbf{R})$ decomposition of LLMs that addresses this gap. Our approach first introduces a novel 3-Block Alternating Direction Method of Multipliers (ADMM) method, termed $\texttt{3BASiL}$, to minimize the layer-wise reconstruction error with convergence guarantees. We then design a transformer-matching ($\texttt{TM}$) refinement step that jointly optimizes the sparse and low-rank components across transformer layers. This step minimizes a novel memory-efficient loss that aligns outputs at the transformer level. Notably, the $\texttt{TM}$ procedure is universal as it can enhance any $(\mathbf{S} + \mathbf{L}\mathbf{R})$ decomposition, including pure sparsity. Our numerical experiments show that $\texttt{3BASiL-TM}$ reduces the WikiText2 perplexity gap to dense LLaMA-8B model by over 30% under a (2:4 Sparse + 64 LR) configuration, compared to prior methods. Moreover, our method achieves over 2.5x faster compression runtime on an A100 GPU compared to SOTA $(\mathbf{S} + \mathbf{L}\mathbf{R})$ method. Our code is available at https://github.com/mazumder-lab/3BASiL.

Deep Learning · Generative Models and Autoencoders

Amin Heyrani Nobari, Lyle Regenwetter, Cyril Picard, Ligong Han, Faez Ahmed

Structural topology optimization (TO) is central to engineering design but remains computationally intensive due to complex physics and hard constraints. Existing deep-learning methods are limited to fixed square grids, a few hand-coded boundary conditions, and post-hoc optimization, preventing general deployment. We introduce Optimize Any Topology (OAT), a foundation-model framework that directly predicts minimum-compliance layouts for arbitrary aspect ratios, resolutions, volume fractions, loads, and fixtures. OAT combines a resolution- and shape-agnostic autoencoder with an implicit neural-field decoder and a conditional latent-diffusion model trained on OpenTO, a new corpus of 2.2 million optimized structures covering 2 million unique boundary-condition configurations. On four public benchmarks and two challenging unseen tests, OAT lowers mean compliance up to 90% relative to the best prior models and delivers sub-1 second inference on a single GPU across resolutions from 64 × 64 to 256 x 256 and aspect ratios as high as 10:1. These results establish OAT as a general, fast, and resolution-free framework for physics-aware topology optimization and provide a large-scale dataset to spur further research in generative modeling for inverse design. Code & data can be found at https://github.com/ahnobari/OptimizeAnyTopology.

Deep Learning · Attention Mechanisms

Yuhong CHOU, Zehao Liu, Rui-Jie Zhu, Xinyi Wan, Tianjian Li, Congying Chu, Qian Liu, Jibin Wu, Zejun MA

Linear attention mechanisms deliver significant advantages for Large Language Models (LLMs) by providing linear computational complexity, enabling efficient processing of ultra-long sequences (e.g., 1M context). However, existing Sequence Parallelism (SP) methods, essential for distributing these workloads across devices, become the primary performance bottleneck due to substantial communication overhead. In this paper, we introduce ZeCO (Zero Communication Overhead) sequence parallelism for linear attention models, a new SP method designed to overcome these limitations and achieve practically end-to-end near-linear scalability for long sequence training. For example, training a model with a 1M sequence length across 64 devices using ZeCO takes roughly the same time as training with an 16k sequence on a single device. At the heart of ZeCO lies All-Scan, a novel collective communication primitive. All-Scan provides each SP rank with precisely the initial operator state it requires while maintaining a minimal communication footprint, effectively eliminating communication overhead. Theoretically, we prove the optimaity of ZeCO, showing that it introduces only negligible time and space overhead. Empirically, we compare the communication costs of different sequence parallelism strategies and demonstrate that All-Scan achieves the fastest communication in SP scenarios. Specifically, on 256 GPUs with an 8M sequence length, ZeCO achieves a 60\% speedup compared to the current state-of-the-art (SOTA) SP method. We believe ZeCO establishes a clear path toward efficiently training next-generation LLMs on previously intractable sequence lengths.

Computer Vision · Vision Models & Multimodal

Yandan Yang, Baoxiong Jia, Shujie Zhang, Siyuan Huang

Indoor scene synthesis has become increasingly important with the rise of Embodied AI, which requires 3D environments that are not only visually realistic but also physically plausible and functionally diverse. While recent approaches have advanced visual fidelity, they often remain constrained to fixed scene categories, lack sufficient object-level detail and physical consistency, and struggle to align with complex user instructions. In this work, we present SceneWeaver, a reflective agentic framework that unifies diverse scene synthesis paradigms through tool-based iterative refinement. At its core, SceneWeaver employs a language model-based planner to select from a suite of extensible scene generation tools, ranging from data-driven generative models to visual- and LLM-based methods, guided by self-evaluation of physical plausibility, visual realism, and semantic alignment with user input. This closed-loop reason-act-reflect design enables the agent to identify semantic inconsistencies, invoke targeted tools, and update the environment over successive iterations. Extensive experiments on both common and open-vocabulary room types demonstrate that \model not only outperforms prior methods on physical, visual, and semantic metrics, but also generalizes effectively to complex scenes with diverse instructions, marking a step toward general-purpose 3D environment generation.

Deep Learning · Generative Models and Autoencoders

Jiahao Yu, Zelei Cheng, Xian Wu, Xinyu Xing

Large language models (LLMs) are increasingly used in various domains, showing impressive potential on various tasks. Recently, reasoning LLMs have been proposed to improve the \textit{reasoning} or \textit{thinking} capabilities of LLMs to solve complex problems. Despite the promising results of reasoning LLMs, enhancing the multi-step reasoning capabilities of LLMs still remains a significant challenge. While existing optimization methods have advanced the LLM reasoning capabilities, they often treat reasoning trajectories as a whole, without considering the underlying critical steps within the trajectory. In this paper, we introduce \textbf{G}uided \textbf{P}ivotal \textbf{O}ptimization (GPO), a novel fine-tuning strategy that dives into the reasoning process to enable more effective improvements. GPO first identifies the `critical step' within a reasoning trajectory - a point that the model must carefully proceed so as to succeed at the problem. We locate the critical step by estimating the advantage function. GPO then resets the policy to the critical step and samples the new rollout and prioritizes learning process on those rollouts. This focus allows the model to learn more effectively from pivotal moments within the reasoning process to improve the reasoning performance. We demonstrate that GPO is not a standalone method, but rather a general strategy that can be integrated with various optimization methods to improve reasoning performance. Besides theoretical analysis, our experiments across challenging reasoning benchmarks show that GPO can consistently and significantly enhances the performance of existing optimization methods, showcasing its effectiveness and generalizability in improving LLM reasoning by concentrating on pivotal moments within the generation process.

Optimization · Optimization and Learning under Uncertainty

Xiang Li, Yunai Li, Huiying Zhong, Lihua Lei, Zhun Deng

Performativity of predictions refers to the phenomenon where prediction-informed decisions influence the very targets they aim to predict—a dynamic commonly observed in policy-making, social sciences, and economics. In this paper, we initiate an end-to-end framework of statistical inference under performativity. Our contributions are twofold. First, we establish a central limit theorem for estimation and inference in the performative setting, enabling standard inferential tasks such as constructing confidence intervals and conducting hypothesis tests in policy-making contexts. Second, we leverage this central limit theorem to study prediction-powered inference (PPI) under performativity. This approach yields more precise estimates and tighter confidence regions for the model parameters (i.e., policies) of interest in performative prediction. We validate the effectiveness of our framework through numerical experiments. To the best of our knowledge, this is the first work to establish a complete statistical inference under performativity, introducing new challenges and inference settings that we believe will provide substantial value to policy-making, statistics, and machine learning.

Theory · Learning Theory

BaoLinh Tran, Van Vu

The matrix recovery (completion) problem, a central problem in data science, involves recovering a matrix $A$ from a relatively small random set of entries. While such a task is generally impossible, it has been shown that one can recover $A$ exactly in polynomial time, with high probability, under three basic and necessary assumptions: (1) the rank of $A$ is very small compared to its dimensions (low rank), (2) $A$ has delocalized singular vectors (incoherence), and (3) the sample size is sufficiently large. Various algorithms address this task, including convex optimization by Candes, Recht, and Tao (2009, 2010), alternating projection by Hardt and Wooters (2014), and low-rank approximation with gradient descent by Keshavan, Montanari, and Oh (2009, 2010). In applications, Candes and Plan (2009) noted that it is more realistic to assume noisy observations. In such cases, the above approaches provide approximate recovery with small root mean square error, which is difficult to convert into exact recovery. Recently, results by Abbe et al. (2017) and Bhardwaj et al. (2023) on approximation in the infinity norm showed that one can recover $A$ even in the noisy case, provided $A$ has bounded precision. However, beyond the three basic assumptions, they either required that the condition number of $A$ be small (2017) or that the gaps between consecutive singular values be large (2023). These additional assumptions conflict, with one requiring singular values to be close together and the other suggesting they should be far apart. It is thus natural to conjecture that neither is necessary. In this paper, we demonstrate that this is indeed the case. We propose a simple algorithm for exact recovery of noisy data, relying solely on the three basic assumptions. The core step of the algorithm is a straightforward truncated singular value decomposition, which is highly efficient. To analyze the algorithm, we prove a new infinity norm version of the classical Davis-Kahan perturbation theorem, improving an earlier result in (2023). Our proof employs a combinatorial contour integration argument and is entirely distinct from all previous approaches.

Deep Learning · Attention Mechanisms

Susav Shrestha, Bradley Settlemyer, Nikoli Dryden, Narasimha Reddy

Accelerating large language model (LLM) inference is critical for real-world deployments requiring high throughput and low latency. Contextual sparsity, where each token dynamically activates only a small subset of the model parameters, shows promise but does not scale to large batch sizes due to union of active neurons quickly approaching dense computation. We introduce Polar Sparsity, highlighting a key shift in sparsity importance from MLP to Attention layers as we scale batch size and sequence length. While MLP layers become more compute-efficient under batching, their sparsity vanishes. In contrast, attention becomes increasingly more expensive at scale, while their head sparsity remains stable and batch-invariant. We develop Selective Head Attention with hardware-efficient, sparsity-aware GPU kernels, delivering up to \(2.2\times\) end-to-end speedups for models like OPT, LLaMA-2 \& 3, Qwen, Mistral across various batch sizes and sequence lengths without compromising accuracy. To our knowledge, this is the first work to demonstrate that contextual sparsity can scale effectively to large batch sizes, delivering substantial inference acceleration with minimal changes, making Polar Sparsity practical for large-scale, high-throughput LLM deployment systems.

Deep Learning · Theory

Hector Pasten, Felipe Urrutia, Hector Orellana, Cristian Buc Calderon, Cristobal Rojas, Alexander Kozachinskiy

Understanding how Transformers work and how they process information is key to the theoretical and empirical advancement of these machines. In this work, we demonstrate the existence of two phenomena in Transformers, namely _isolation_ and _continuity_. Both of these phenomena hinder Transformers to learn even simple pattern sequences. Isolation expresses that any learnable sequence must be isolated from another learnable sequence, and hence some sequences cannot be learned by a single Transformer at the same time. Continuity entails that an attractor basin forms around a learned sequence, such that any sequence falling in that basin will collapse towards the learned sequence. Here, we mathematically prove these phenomena emerge in all Transformers that use compact positional encoding, and design rigorous experiments, demonstrating that the theoretical limitations we shed light on occur on the practical scale.

Yulei Qin, Gang Li, Zongyi Li, Zihan Xu, Yuchen Shi, Zhekai Lin, Xiao Cui, Ke Li, Xing Sun

Existing large language models (LLMs) face challenges of following complex instructions, especially when multiple constraints are present and organized in paralleling, chaining, and branching structures. One intuitive solution, namely chain-of-thought (CoT), is expected to universally improve capabilities of LLMs. However, we find that the vanilla CoT exerts a negative impact on performance due to its superficial reasoning pattern of simply paraphrasing the instructions. It fails to peel back the compositions of constraints for identifying their relationship across hierarchies of types and dimensions. To this end, we propose RAIF, a systematic method to boost LLMs in dealing with complex instructions via incentivizing reasoning for test-time compute scaling. First, we stem from the decomposition of complex instructions under existing taxonomies and propose a reproducible data acquisition method. Second, we exploit reinforcement learning (RL) with verifiable rule-centric reward signals to cultivate reasoning specifically for instruction following. We address the shallow, non-essential nature of reasoning under complex instructions via sample-wise contrast for superior CoT enforcement. We also exploit behavior cloning of experts to facilitate steady distribution shift from fast-thinking LLMs to skillful reasoners. Extensive evaluations on seven comprehensive benchmarks confirm the validity of the proposed method, where a 1.5B LLM achieves 11.74% gains with performance comparable to a 8B LLM. Evaluation on OOD constraints also confirms the generalizability of our RAIF.

Deep Learning · Generative Models and Autoencoders

Yifei Liu, Li Lyna Zhang, Yi Zhu, Bingcheng Dong, Xudong Zhou, Ning Shang, Fan Yang, Cheng Li, Mao Yang

Advancing code reasoning in large language models (LLMs) is fundamentally limited by the scarcity of high-difficulty datasets, especially those with verifiable input-output test cases necessary for rigorous solution validation at scale. We introduce rStar-Coder, which significantly improves LLM code reasoning capabilities by constructing a large-scale, verified dataset of 418K competition-level code problems, 580K long-reasoning solutions along with rich test cases of varying difficulty. This is achieved through three core contributions: (1) we curate competitive programming code problems and solutions to synthesize new, solvable problems; (2) we introduce a reliable input-output test case synthesis pipeline that decouples the generation into a three-step input generation method and a mutual verification mechanism for effective output labeling; (3) we augment problems with high-quality, test-case-verified long-reasoning solutions. Extensive experiments on Qwen models (1.5B-14B) across various code reasoning benchmarks demonstrate the superiority of rStar-Coder dataset, achieving leading performance comparable to frontier reasoning LLMs with significantly smaller model sizes. On LiveCodeBench, rStar-Coder improves Qwen2.5-7B from 17.4% to an impressive 57.3%, and Qwen2.5-14B from 23.3% to 62.5%, surpassing o3-mini (low) by 3.1%. On the more challenging USA Computing Olympiad, our 7B model achieves an average pass@1 accuracy of 16.15%, outperforming the frontier-level QWQ-32B. rStar-Coder dataset is publicly available at https://huggingface.co/datasets/microsoft/rStar-Coder.

Deep Learning · Generative Models and Autoencoders

Bartlomiej Sobieski, Matthew Tivnan, Yuang Wang, Siyeop yoon, Pengfei Jin, Dufan Wu, Quanzheng Li, Przemyslaw Biecek

Solving inverse problems—recovering signals from incomplete or noisy measurements—is fundamental in science and engineering. Score-based generative models (SGMs) have recently emerged as a powerful framework for this task. Two main paradigms have formed: unsupervised approaches that adapt pretrained generative models to inverse problems, and supervised bridge methods that train stochastic processes conditioned on paired clean and corrupted data. While the former typically assume knowledge of the measurement model, the latter have largely overlooked this structural information. We introduce System-embedded Diffusion Bridge Models (SDBs), a new class of supervised bridge methods that explicitly embed the known linear measurement system into the coefficients of a matrix-valued SDE. This principled integration yields consistent improvements across diverse linear inverse problems and demonstrates robust generalization under system misspecification between training and deployment, offering a promising solution to real-world applications.

Computer Vision · Everything Else

Zhiqiang Yan, Jianhao Jiao, Zhengxue Wang, Gim Hee Lee

Depth completion in dynamic scenes poses significant challenges due to rapid ego-motion and object motion, which can severely degrade the quality of input modalities such as RGB images and LiDAR measurements. Conventional RGB-D sensors often struggle to align precisely and capture reliable depth under such conditions. In contrast, event cameras with their high temporal resolution and sensitivity to motion at the pixel level provide complementary cues that are beneficial in dynamic environments. To this end, we propose EventDC, the first event-driven depth completion framework. It consists of two key components: Event-Modulated Alignment (EMA) and Local Depth Filtering (LDF). Both modules adaptively learn the two fundamental components of convolution operations: offsets and weights conditioned on motion-sensitive event streams. In the encoder, EMA leverages events to modulate the sampling positions of RGB-D features to achieve pixel redistribution for improved alignment and fusion. In the decoder, LDF refines depth estimations around moving objects by learning motion-aware masks from events. Additionally, EventDC incorporates two loss terms to further benefit global alignment and enhance local depth recovery. Moreover, we establish the first benchmark for event-based depth completion comprising one real-world and two synthetic datasets to facilitate future research. Extensive experiments on this benchmark demonstrate the superiority of our EventDC. [Project page](https://yanzq95.github.io/projectpage/EventDC/index.html).

Computer Vision · 3D Rendering & Reconstruction

Jonathan Schmidt, Simon Giebenhain, Matthias Niessner

We introduce *BecomingLit*, a novel method for reconstructing relightable, high-resolution head avatars that can be rendered from novel viewpoints at interactive rates. Therefore, we propose a new low-cost light stage capture setup, tailored specifically towards capturing faces. Using this setup, we collect a novel dataset consisting of diverse multi-view sequences of numerous subjects under varying illumination conditions and facial expressions. By leveraging our new dataset, we introduce a new relightable avatar representation based on 3D Gaussian primitives that we animate with a parametric head model and an expression-dependent dynamics module. We propose a new hybrid neural shading approach, combining a neural diffuse BRDF with an analytical specular term. Our method reconstructs disentangled materials from our dynamic light stage recordings and enables all-frequency relighting of our avatars with both point lights and environment maps. In addition, our avatars can easily be animated and controlled from monocular videos. We validate our approach in extensive experiments on our dataset, where we consistently outperform existing state-of-the-art methods in relighting and reenactment by a significant margin.

General Machine Learning · Representation Learning

Jiaxin Wu, Chenglong Pang, Guangxiong Chen, Jie Zhao

Graph Convolutional Networks (GCNs) based on linear aggregation have been widely applied across various domains due to their exceptional performance. To enhance performance, these networks often utilize the graph Laplacian norm to suppress the propagation of information from first-order neighbors. However, this approach may dilute valuable interaction information and make the model slowly learn sparse interaction relationships from neighbors, which increases training time and negatively affects performance. To address these issues, we introduce BoostGCN, a novel linear GCN model that focuses on amplifying significant interactions with first-order neighbors, which enables the model to accurately and quickly capture significant relationships. BoostGCN has relatively fixed parameters, making it user-friendly. Experiments on four real-world datasets demonstrate that BoostGCN outperforms existing state-of-the-art GCN models in both performance and efficiency.

Deep Learning · Generative Models and Autoencoders

Dexiong Chen, Markus Krimmel, Karsten Borgwardt

We introduce AutoGraph, a scalable autoregressive model for attributed graph generation using decoder-only transformers. By flattening graphs into random sequences of tokens through a reversible process, AutoGraph enables modeling graphs as sequences without relying on additional node features that are expensive to compute, in contrast to diffusion-based approaches. This results in sampling complexity and sequence lengths that scale optimally linearly with the number of edges, making it scalable and efficient for large, sparse graphs. A key success factor of AutoGraph is that its sequence prefixes represent induced subgraphs, creating a direct link to sub-sentences in language modeling. Empirically, AutoGraph achieves state-of-the-art performance on synthetic and molecular benchmarks, with up to 100x faster generation and 3x faster training than leading diffusion models. It also supports substructure-conditioned generation without fine-tuning and shows promising transferability, bridging language modeling and graph generation to lay the groundwork for graph foundation models. Our code is available at https://github.com/BorgwardtLab/AutoGraph.

General Machine Learning · Transfer, Multitask and Meta-learning

Ke Wang, Yiming QIN, Nikolaos Dimitriadis, Alessandro Favero, Pascal Frossard

Language models deployed in real-world systems often require post-hoc updates to incorporate new or corrected knowledge. However, editing such models efficiently and reliably—without retraining or forgetting previous information—remains a major challenge. Existing methods for lifelong model editing either compromise generalization, interfere with past edits, or fail to scale to long editing sequences. We propose MEMOIR, a novel scalable framework that injects knowledge through a residual memory, i.e., a dedicated parameter module, while preserving the core capabilities of the pre-trained model. By sparsifying input activations through data-dependent masks, MEMOIR confines each edit to a distinct subset of the memory parameters, minimizing interference among edits. At inference, it identifies relevant edits by comparing the sparse activation patterns of new queries to those stored during editing. This enables generalization to rephrased queries by activating only the relevant knowledge while suppressing unnecessary memory activation for unrelated prompts. Experiments on question answering, hallucination correction, and out-of-distribution generalization benchmarks across LLaMA-3 and Mistral demonstrate that MEMOIR achieves state-of-the-art performance across reliability, generalization, and locality metrics, scaling to thousands of sequential edits with minimal forgetting.

Computer Vision · 3D Rendering & Reconstruction

Qing Li, Huifang Feng, Xun Gong, Yu-Shen Liu

3D Gaussian Splatting has recently emerged as an efficient solution for high-quality and real-time novel view synthesis. However, its capability for accurate surface reconstruction remains underexplored. Due to the discrete and unstructured nature of Gaussians, supervision based solely on image rendering loss often leads to inaccurate geometry and inconsistent multi-view alignment. In this work, we propose a novel method that enhances the geometric representation of 3D Gaussians through view alignment (VA). Specifically, we incorporate edge-aware image cues into the rendering loss to improve surface boundary delineation. To enforce geometric consistency across views, we introduce a visibility-aware photometric alignment loss that models occlusions and encourages accurate spatial relationships among Gaussians. To further mitigate ambiguities caused by lighting variations, we incorporate normal-based constraints to refine the spatial orientation of Gaussians and improve local surface estimation. Additionally, we leverage deep image feature embeddings to enforce cross-view consistency, enhancing the robustness of the learned geometry under varying viewpoints and illumination. Extensive experiments on standard benchmarks demonstrate that our method achieves state-of-the-art performance in both surface reconstruction and novel view synthesis. The source code is available at https://github.com/LeoQLi/VA-GS.

Deep Learning · Everything Else

Howe Tissue, Venus Wang, Lu Wang

We find that the cross-entropy loss curves of neural language models empirically adhere to a scaling law with learning rate (LR) annealing over training steps: $$L(s) = L_0 + A\cdot S_1^{-\alpha} - C\cdot S_2,$$ where $L(s)$ is the validation loss at step $s$, $S_1$ is the area under the LR curve, $S_2$ is the LR annealing area, and $L_0$, $A$, $C$, $\alpha$ are constant parameters. This formulation accounts for two main effects: (1) power-law scaling over data size, and (2) the additional loss reduction during LR annealing. Unlike previous studies that only fit losses at final steps, our formulation captures the entire training curve, allowing for parameter fitting using losses from any training step. Applying the scaling law with LR annealing and fitting only one or two training curves, we can accurately predict the loss at any given step under any learning rate scheduler (LRS). This approach significantly reduces computational cost in formulating scaling laws while providing more accuracy and expressiveness. Extensive experiments demonstrate that our findings hold across a range of hyper-parameters and model architectures and can extend to scaling effect of model sizes. Moreover, our formulation provides accurate theoretical insights into empirical results observed in numerous previous studies, particularly those focusing on LR schedule and annealing. We believe that this work is promising to enhance the understanding of LLM training dynamics while democratizing scaling laws, and it is helpful to guide both research and industrial participants in refining training strategies for further LLMs.

Computer Vision · Classification and Understanding

Hao-Ran Yang, Xiaohui Chen, Chuan-Xian Ren

Aiming to generalize the well-trained gaze estimation model to new target domains, Cross-domain Gaze Estimation (CDGE) is developed for real-world application scenarios. Existing CDGE methods typically extract the domain-invariant features to mitigate domain shift in feature space, which is proved insufficient by Generalized Label Shift (GLS) theory. In this paper, we introduce a novel GLS perspective to CDGE and modelize the cross-domain problem by label and conditional shift problem. A GLS correction framework is presented and a feasible realization is proposed, in which a importance reweighting strategy based on truncated Gaussian distribution is introduced to overcome the continuity challenges in label shift correction. To embed the reweighted source distribution to conditional invariant learning, we further derive a probability-aware estimation of conditional operator discrepancy. Extensive experiments on standard CDGE tasks with different backbone models validate the superior generalization capability across domain and applicability on various models of proposed method.