Weight Pruning Research Articles

Novel Pruning Techniques in Convolutional Neural Networks

Deep Learning allows us to build powerful models to solve problems like image classification, time series prediction, natural language processing, etc. This is achieved at the cost of huge amounts of storage and processing requirements which are sometimes not possible in machines with limited resources. In this paper, we compare different methods which tackle this problem with network pruning. Selected few pruning methodologies from the deep learning literature were implemented to display their results. Modern neural architectures have a combination of different layers like convolutional layers, pooling layers, dense layers, etc. We compare pruning techniques for dense layers (such as unit/neuron pruning, and weight Pruning), and convolutional layers as well (using L1 norm, taylor expansion of loss to determine importance of convolutional filters, and Variable Importance in Projection using Partial Least Squares) for the image classification task. This study aims to ease the overhead in terms of optimization of the model for academic, as well as commercial, use of deep neural networks.

Harmonious Coexistence of Structured Weight Pruning and Ternarization for Deep Neural Networks

Deep convolutional neural network (DNN) has demonstrated phenomenal success and been widely used in many computer vision tasks. However, its enormous model size and high computing complexity prohibits its wide deployment into resource limited embedded system, such as FPGA and mGPU. As the two most widely adopted model compression techniques, weight pruning and quantization compress DNN model through introducing weight sparsity (i.e., forcing partial weights as zeros) and quantizing weights into limited bit-width values, respectively. Although there are works attempting to combine the weight pruning and quantization, we still observe disharmony between weight pruning and quantization, especially when more aggressive compression schemes (e.g., Structured pruning and low bit-width quantization) are used. In this work, taking FPGA as the test computing platform and Processing Elements (PE) as the basic parallel computing unit, we first propose a PE-wise structured pruning scheme, which introduces weight sparsification with considering of the architecture of PE. In addition, we integrate it with an optimized weight ternarization approach which quantizes weights into ternary values ({-1,0,+1}), thus converting the dominant convolution operations in DNN from multiplication-and-accumulation (MAC) to addition-only, as well as compressing the original model (from 32-bit floating point to 2-bit ternary representation) by at least 16 times. Then, we investigate and solve the coexistence issue between PE-wise Structured pruning and ternarization, through proposing a Weight Penalty Clipping (WPC) technique with self-adapting threshold. Our experiment shows that the fusion of our proposed techniques can achieve the best state-of-the-art ∼21× PE-wise structured compression rate with merely 1.74%/0.94% (top-1/top-5) accuracy degradation of ResNet-18 on ImageNet dataset.

AutoCompress: An Automatic DNN Structured Pruning Framework for Ultra-High Compression Rates

Structured weight pruning is a representative model compression technique of DNNs to reduce the storage and computation requirements and accelerate inference. An automatic hyperparameter determination process is necessary due to the large number of flexible hyperparameters. This work proposes AutoCompress, an automatic structured pruning framework with the following key performance improvements: (i) effectively incorporate the combination of structured pruning schemes in the automatic process; (ii) adopt the state-of-art ADMM-based structured weight pruning as the core algorithm, and propose an innovative additional purification step for further weight reduction without accuracy loss; and (iii) develop effective heuristic search method enhanced by experience-based guided search, replacing the prior deep reinforcement learning technique which has underlying incompatibility with the target pruning problem. Extensive experiments on CIFAR-10 and ImageNet datasets demonstrate that AutoCompress is the key to achieve ultra-high pruning rates on the number of weights and FLOPs that cannot be achieved before. As an example, AutoCompress outperforms the prior work on automatic model compression by up to 33× in pruning rate (120× reduction in the actual parameter count) under the same accuracy. Significant inference speedup has been observed from the AutoCompress framework on actual measurements on smartphone. We release models of this work at anonymous link: http://bit.ly/2VZ63dS.

PCONV: The Missing but Desirable Sparsity in DNN Weight Pruning for Real-Time Execution on Mobile Devices

Model compression techniques on Deep Neural Network (DNN) have been widely acknowledged as an effective way to achieve acceleration on a variety of platforms, and DNN weight pruning is a straightforward and effective method. There are currently two mainstreams of pruning methods representing two extremes of pruning regularity: non-structured, fine-grained pruning can achieve high sparsity and accuracy, but is not hardware friendly; structured, coarse-grained pruning exploits hardware-efficient structures in pruning, but suffers from accuracy drop when the pruning rate is high. In this paper, we introduce PCONV, comprising a new sparsity dimension, – fine-grained pruning patterns inside the coarse-grained structures. PCONV comprises two types of sparsities, Sparse Convolution Patterns (SCP) which is generated from intra-convolution kernel pruning and connectivity sparsity generated from inter-convolution kernel pruning. Essentially, SCP enhances accuracy due to its special vision properties, and connectivity sparsity increases pruning rate while maintaining balanced workload on filter computation. To deploy PCONV, we develop a novel compiler-assisted DNN inference framework and execute PCONV models in real-time without accuracy compromise, which cannot be achieved in prior work. Our experimental results show that, PCONV outperforms three state-of-art end-to-end DNN frameworks, TensorFlow-Lite, TVM, and Alibaba Mobile Neural Network with speedup up to 39.2 ×, 11.4 ×, and 6.3 ×, respectively, with no accuracy loss. Mobile devices can achieve real-time inference on large-scale DNNs.

Cluster Pruning: An Efficient Filter Pruning Method for Edge AI Vision Applications

Even though the Convolutional Neural Networks (CNN) has shown superior results in the field of computer vision, it is still a challenging task to implement computer vision algorithms in real-time at the edge, especially using a low-cost IoT device due to high memory consumption and computation complexities in a CNN. Network compression methodologies such as weight pruning, filter pruning, and quantization are used to overcome the above mentioned problem. Even though filter pruning methodology has shown better performances compared to other techniques, irregularity of the number of filters pruned across different layers of a CNN might not comply with majority of the neural computing hardware architectures. In this paper, a novel greedy approach called cluster pruning has been proposed, which provides a structured way of removing filters in a CNN by considering the importance of filters and the underlying hardware architecture. The proposed methodology is compared with the conventional filter pruning algorithm on Pascal-VOC open dataset, and Head-Counting dataset, which is our own dataset developed to detect and count people entering a room. We benchmark our proposed method on three hardware architectures, namely CPU, GPU, and Intel Movidius Neural Computer Stick (NCS) using the popular SSD-MobileNet and SSD-SqueezeNet neural network architectures used for edge-AI vision applications. Results demonstrate that our method outperforms the conventional filter pruning methodology, using both datasets on above mentioned hardware architectures. Furthermore, a low cost IoT hardware setup consisting of an Intel Movidius-NCS is proposed to deploy an edge-AI application using our proposed pruning methodology.

3D Capsule Networks for Object Classification With Weight Pruning

The proliferation of 3D sensors, due to the increased demand for 3D data, induced the 3D computer vision research in the last decade, and 3D data processing has gained a lot of interest. As in many other applications in computer vision, deep learning-based methods were quickly applied to 3D data classification and have become the state-of-the-art in this area. More recently, capsule networks , which are novel neural structures, have been introduced to enhance the ability of neural networks to better capture the parts-relationship, which yields more accurate classification with less training data. Moreover, deploying deep machine learning models on mobile platforms requires the models to be optimized due to limited memory and computational constraints. In this work, we propose methods to boost the accuracies of a standard 3D CNN-based and a Capsule Network-based classifier, help the training to better generalize the data distribution with limited data, and optimize the models for resource-constrained environments, such as mobile platforms. We also show that the introduction of capsules to 3D object classification pipeline improves the classification performance with limited training data, while a specifically optimized weight pruning method keeps the model compact enough for mobile deployment. Our broad spectrum of experiments show that proposed methods improve the performance of the base model while significantly reducing the memory and computation requirements.

Learning Sparse Low-Precision Neural Networks With Learnable Regularization

We consider learning deep neural networks (DNNs) that consist of low-precision weights and activations for efficient inference of fixed-point operations. In training low-precision networks, gradient descent in the backward pass is performed with high-precision weights while quantized low-precision weights and activations are used in the forward pass to calculate the loss function for training. Thus, the gradient descent becomes suboptimal, and accuracy loss follows. In order to reduce the mismatch in the forward and backward passes, we utilize mean squared quantization error (MSQE) regularization. In particular, we propose using a learnable regularization coefficient with the MSQE regularizer to reinforce the convergence of high-precision weights to their quantized values. We also investigate how partial L2 regularization can be employed for weight pruning in a similar manner. Finally, combining weight pruning, quantization, and entropy coding, we establish a low-precision DNN compression pipeline. In our experiments, the proposed method yields low-precision MobileNet and ShuffleNet models on ImageNet classification with the state-of-the-art compression ratios of 7.13 and 6.79, respectively. Moreover, we examine our method for image super resolution networks to produce 8-bit low-precision models at negligible performance loss.

Design and Application of Transmission Line Intelligent Monitoring System

This paper presents a design method of intelligent monitoring system for transmission lines based on artificial intelligence technology. In this design method, a low-power artificial intelligence chip - LieYing A101 is used to design an intelligent recognition module to realize real-time target recognition on a terminal device. In order to solve the problem that the original image and the input image resolution of the intelligent recognition module do not match, this paper uses a sliding window and convolutional neural network design method, which solves the image resolution mismatch problem and improves the recognition accuracy. Finally, for the problem of excessive network model size, feature channel weight pruning and 8-bit quantization methods are used to compress the network model to less than 10M, and the recognition accuracy is not sharply reduced. After the test set test and actual scene use, the external force destruction target recognition accuracy of the transmission line channel is high; this meets the application needs of customers.

Memory-Reduced Network Stacking for Edge-Level CNN Architecture With Structured Weight Pruning

This paper presents a novel stacking and multi-level indexing scheme for convolutional neural networks (CNNs) used in energy-limited edge-level systems. Basically, the proposed scheme offers multiple accuracy modes by adopting a structured weight pruning method that enables a CNN to be trained once with multiple pruning ratios and thereby allows for adaptive energy-accuracy trade-offs. The memory overhead required to store several different networks is kept to a minimum by adopting a novel method for including smaller lower-accuracy networks as subnetworks of larger higher-accuracy networks and by using a unique multi-level indexing scheme that can effectively store compressed weight data for the proposed stacked-CNN architecture. Experimental results show that the proposed method successfully reduces the memory footprint by up to 33% when compared to a baseline CNN architecture. An FPGA-based multi-mode CNN accelerator that implements the proposed scheme has been designed. Energy usage analysis with a case study shows that the inference energy required for on-device CNN processing can be reduced by up to 1.94 times over the baseline design.

Deepprune: Learning Efficient and Interpretable Convolutional Networks Through Weight Pruning for Predicting DNA-Protein Binding.

Convolutional neural network (CNN) based methods have outperformed conventional machine learning methods in predicting the binding preference of DNA-protein binding. Although studies in the past have shown that more convolutional kernels help to achieve better performance, visualization of the model can be obscured by the use of many kernels, resulting in overfitting and reduced interpretation because the number of motifs in true models is limited. Therefore, we aim to arrive at high performance, but with limited kernel numbers, in CNN-based models for motif inference. We herein present Deepprune, a novel deep learning framework, which prunes the weights in the dense layer and fine-tunes iteratively. These two steps enable the training of CNN-based models with limited kernel numbers, allowing easy interpretation of the learned model. We demonstrate that Deepprune significantly improves motif inference performance for the simulated datasets. Furthermore, we show that Deepprune outperforms the baseline with limited kernel numbers when inferring DNA-binding sites from ChIP-seq data.

Fast Convolutional Neural Networks in Low Density FPGAs Using Zero-Skipping and Weight Pruning

Edge devices are becoming smarter with the integration of machine learning methods, such as deep learning, and are therefore used in many application domains where decisions have to be made without human intervention. Deep learning and, in particular, convolutional neural networks (CNN) are more efficient than previous algorithms for several computer vision applications such as security and surveillance, where image and video analysis are required. This better efficiency comes with a cost of high computation and memory requirements. Hence, running CNNs in embedded computing devices is a challenge for both algorithm and hardware designers. New processing devices, dedicated system architectures and optimization of the networks have been researched to deal with these computation requirements. In this paper, we improve the inference execution times of CNNs in low density FPGAs (Field-Programmable Gate Arrays) using fixed-point arithmetic, zero-skipping and weight pruning. The developed architecture supports the execution of large CNNs in FPGA devices with reduced on-chip memory and computing resources. With the proposed architecture, it is possible to infer an image in AlexNet in 2.9 ms in a ZYNQ7020 and 1.0 ms in a ZYNQ7045 with less than 1% accuracy degradation. These results improve previous state-of-the-art architectures for CNN inference.

Combining handcrafted features with latent variables in machine learning for prediction of radiation-induced lung damage.

There has been burgeoning interest in applying machine learning methods for predicting radiotherapy outcomes. However, the imbalanced ratio of a large number of variables to a limited sample size in radiation oncology constitutes a major challenge. Therefore, dimensionality reduction methods can be a key to success. The study investigates and contrasts the application of traditional machine learning methods and deep learning approaches for outcome modeling in radiotherapy. In particular, new joint architectures based on variational autoencoder (VAE) for dimensionality reduction are presented and their application is demonstrated for the prediction of lung radiation pneumonitis (RP) from a large-scale heterogeneous dataset. A large-scale heterogeneous dataset containing a pool of 230 variables including clinical factors (e.g., dose, KPS, stage) and biomarkers (e.g., single nucleotide polymorphisms (SNPs), cytokines, and micro-RNAs) in a population of 106 nonsmall cell lung cancer (NSCLC) patients who received radiotherapy was used for modeling RP. Twenty-two patients had grade 2 or higher RP. Four methods were investigated, including feature selection (case A) and feature extraction (case B) with traditional machine learning methods, a VAE-MLP joint architecture (case C) with deep learning and lastly, the combination of feature selection and joint architecture (case D). For feature selection, Random forest (RF), Support Vector Machine (SVM), and multilayer perceptron (MLP) were implemented to select relevant features. Specifically, each method was run for multiple times to rank features within several cross-validated (CV) resampled sets. A collection of ranking lists were then aggregated by top 5% and Kemeny graph methods to identify the final ranking for prediction. A synthetic minority oversampling technique was applied to correct for class imbalance during this process. For deep learning, a VAE-MLP joint architecture where a VAE aimed for dimensionality reduction and an MLP aimed for classification was developed. In this architecture, reconstruction loss and prediction loss were combined into a single loss function to realize simultaneous training and weights were assigned to different classes to mitigate class imbalance. To evaluate the prediction performance and conduct comparisons, the area under receiver operating characteristic curves (AUCs) were performed for nested CVs for both handcrafted feature selections and the deep learning approach. The significance of differences in AUCs was assessed using the DeLong test of U-statistics. An MLP-based method using weight pruning (WP) feature selection yielded the best performance among the different hand-crafted feature selection methods (case A), reaching an AUC of 0.804 (95% CI: 0.761-0.823) with 29 top features. A VAE-MLP joint architecture (case C) achieved a comparable but slightly lower AUC of 0.781 (95% CI: 0.737-0.808) with the size of latent dimension being 2. The combination of handcrafted features (case A) and latent representation (case D) achieved a significant AUC improvement of 0.831 (95% CI: 0.805-0.863) with 22 features (P-value=0.000642 compared with handcrafted features only (Case A) and P-value=0.000453 compared to VAE alone (Case C)) with an MLP classifier. The potential for combination of traditional machine learning methods and deep learning VAE techniques has been demonstrated for dealing with limited datasets in modeling radiotherapy toxicities. Specifically, latent variables from a VAE-MLP joint architecture are able to complement handcrafted features for the prediction of RP and improve prediction over either method alone.

Thinning of convolutional neural network with mixed pruning

Deep learning has achieved state-of-the-art performance in accuracy of many computer vision tasks. However, convolutional neural network is difficult to deploy on resource constrained devices due to their limited computation power and memory space. Thus, it is necessary to prune the redundant weights and filters rationally and effectively. Considering that the pruned model still exists, redundancy after weight pruning or filter pruning alone, a method of combining weight pruning and filter pruning is proposed. First, filter pruning is performed, which is to remove filters with least importance and using fine-tuning to recover the model's accuracy. Then, all connection weights below a threshold are set to zero. Finally, the pruned model obtained by the first two steps is fine-tuned to recover its predictive accuracy. Experiments on MNIST and CIFAR-10 datasets demonstrate that the proposed approach is effective and feasible. Compared with only weight pruning or filter pruning, the mixed pruning can achieve higher compression ratio of the model parameters. For LeNet-5, the proposed approach can achieve a compression rate of 13.01×, with 1% drop in accuracy. For VGG-16, it can achieve a compression rate of 19.20×, incurring 1.56% accuracy loss.

Deep Neural Network Compression by In-Parallel Pruning-Quantization.

Deep neural networks enable state-of-the-art accuracy on visual recognition tasks such as image classification and object detection. However, modern networks contain millions of learned connections, and the current trend is towards deeper and more densely connected architectures. This poses a challenge to the deployment of state-of-the-art networks on resource-constrained systems, such as smartphones or mobile robots. In general, a more efficient utilization of computation resources would assist in deployment scenarios from embedded platforms to computing clusters running ensembles of networks. In this paper, we propose a deep network compression algorithm that performs weight pruning and quantization jointly, and in parallel with fine-tuning. Our approach takes advantage of the complementary nature of pruning and quantization and recovers from premature pruning errors, which is not possible with two-stage approaches. In experiments on ImageNet, CLIP-Q (Compression Learning by In-Parallel Pruning-Quantization) improves the state-of-the-art in network compression on AlexNet, VGGNet, GoogLeNet, and ResNet. We additionally demonstrate that CLIP-Q is complementary to efficient network architecture design by compressing MobileNet and ShuffleNet, and that CLIP-Q generalizes beyond convolutional networks by compressing a memory network for visual question answering.

Energy Efficient Neural Computing: A Study of Cross-Layer Approximations

Deep neural networks (DNNs) have emerged as the state-of-the-art technique in a wide range of machine learning tasks for analytics and computer vision in the next generation of embedded (mobile, IoT, and wearable) devices. Despite their success, they suffer from high energy requirements. In recent years, the inherent error resiliency of DNNs has been exploited by introducing approximations at either the algorithmic or the hardware levels (individually) to obtain energy savings while incurring tolerable accuracy degradation. However, there is a need for investigating the overall energy-accuracy trade-offs arising from the introduction of approximations at different levels in complex DNNs. We perform a comprehensive analysis to determine the effectiveness of cross-layer approximations for the energy-efficient realization of large-scale DNNs. The approximations considered are as follows: 1) use of lower complexity networks (containing lesser number of layers and/or neurons per layer); 2) pruning of synaptic weights; 3) approximate multiplication operation in the neuronal multiply-and-accumulate computation; and 4) approximate write/read operations to/from the synaptic memory. Our experiments on recognition benchmarks (MNIST and CIFAR10) show that cross-layer approximation provides substantial improvements in energy efficiency for different accuracy/quality requirements. Furthermore, we propose a synergistic framework for combining the approximation techniques to achieve maximal energy benefits from approximate DNNs.

Statistically Based Weight Pruning in Feed-Forward Neural Networks

A statistically-based algorithm for pruning weights from feed-forward networks is presented. This algorithm relies upon the Generalized Wald and t-test statistics to determine which weights to remove from the network. Because both of these tests use the exact Hessian matrix, an algorithm for learning the exact Hessian matrix for a feed-forward neural network using a single backward pass through the data is presented when the L2 norm is minimized in the energy function. The pruning algorithm is then applied in two simulations: The first simulation investigates the relationship between neural networks and linear regression (Ordinary Least Squares), and the weight covariance matrix is found to be asymptotically equivalent to the White (1980) standard error corrections for heterogeneity of variance. The final simulation applies the algorithm to a network solving the sunspot data and compares the results to those found in the literature, with mixed results.

A Computationally Efficient Weight Pruning Algorithm for Artificial Neural Network Classifiers

A novel technique is being proposed to prune the weights of artificial neural networks (ANNs) while training with backpropagation algorithm. Iterative update of weights through gradient descent mechanism does not guarantee convergence in a specified number of epochs. Pruning of non-relevant weights not only reduces the computational complexity but also improves the classification performance. This algorithm first defines the “relevance” of initialized weights in a statistical sense by introducing a coefficient of dominance for each weight converging on a hidden node and subsequently employing the concept of complexity penalty. Based upon complexity penalty for each weight, a decision has been taken to either prune or retain the weight. It has been shown analytically that a weight with higher complexity penalty has a high degree of Fisher information which further implies its ability to capture the variations in the input set for better classification. Simulation experiments performed with five benchmark data sets reveal that ANNs trained after being pruned using the proposed technique exhibit higher convergence, lower execution time and higher success rate in the test phase and yields substantial reduction in computational resources. For complex architectures, early convergence was found to be directly correlated with percentage of weights pruned. The efficacy of the technique has been validated on several benchmark datasets having large diversity of attributes.

Scalpel

As the size of Deep Neural Networks (DNNs) continues to grow to increase accuracy and solve more complex problems, their energy footprint also scales. Weight pruning reduces DNN model size and the computation by removing redundant weights. However, we implemented weight pruning for several popular networks on a variety of hardware platforms and observed surprising results. For many networks, the network sparsity caused by weight pruning will actually hurt the overall performance despite large reductions in the model size and required multiply-accumulate operations. Also, encoding the sparse format of pruned networks incurs additional storage space overhead. To overcome these challenges, we propose Scalpel that customizes DNN pruning to the underlying hardware by matching the pruned network structure to the data-parallel hardware organization. Scalpel consists of two techniques: SIMD-aware weight pruning and node pruning. For low-parallelism hardware (e.g., microcontroller), SIMD-aware weight pruning maintains weights in aligned fixed-size groups to fully utilize the SIMD units. For high-parallelism hardware (e.g., GPU), node pruning removes redundant nodes, not redundant weights, thereby reducing computation without sacrificing the dense matrix format. For hardware with moderate parallelism (e.g., desktop CPU), SIMD-aware weight pruning and node pruning are synergistically applied together. Across the microcontroller, CPU and GPU, Scalpel achieves mean speedups of 3.54x, 2.61x, and 1.25x while reducing the model sizes by 88%, 82%, and 53%. In comparison, traditional weight pruning achieves mean speedups of 1.90x, 1.06x, 0.41x across the three platforms.

Extreme learning machine based approach for diagnosis and analysis of breast cancer

Due to the heterogeneous and complex nature of clinical data, the need to use sophisticated diagnosis techniques has increased significantly in recent years. The proposed approach for diagnosis of breast cancer exploits the potential of an extreme learning machine (ELM) and analyzes its performance after classification into benign and malignant cases. To optimize the ELM network in terms of computation time and memory resources, weight pruning is used without performance compromise. Using real data sets, numerical experiments have been conducted. With an accuracy of 93%, the optimum numbers of node layers for breast cancer diagnosis has been found to be 20. Comparative results demonstrate over-performance of the proposed ELM approach.

Cropload Management of ‘Vidal blanc’ Improves Primary Bud Cold Hardiness and Maintains Berry Composition in the Lower Midwestern United States

Canopy architecture, yield components, berry composition, pruning weight, Ravaz Index, and midwinter primary bud cold hardiness of own-rooted ‘Vidal blanc’ (Vitis vinifera × Vitis rupestris) were measured in response to balanced pruning formula treatments of 20, 30, or 40 nodes retained for the first 454 g of dormant pruning weight and an additional 10 nodes for each additional 454 g and three cluster thinning levels of one, two, and two+ clusters per shoot in 2006 and 2007. Although the pruning formula affected the distance between shoots along the canopy, and the number of count shoots per hectare, the canopy leaf layer numbers were unaffected in either year. Application of the pruning formula did not affect components of yield in either year. However, the number of clusters and yield per vine were affected by cluster thinning treatments where they increased linearly with the decrease in its severity, explaining 73% and 77% of total variance in yield in 2006 and 2007, respectively. Pruning formula or cluster thinning did not affect berry composition substantially. Cluster thinning improved the percentage of mature nodes on shoots before a killing frost in both years. Cluster thinning to one or two clusters per shoot also improved the lethal temperature killing 50% of the primary buds compared with no cluster thinning in both years of the study. Mature wood weight and total pruning weight displayed a quadratic response to cluster thinning where two clusters per shoot had the greatest weight for both, whereas pruning formula had no effect on pruning weight. Optimum fruit weight–pruning weight ratio was achieved with the 30 + 10 pruning formula and two clusters per shoot cluster thinning treatments in both years of the study. The results of this study provide valuable information for growers of interspecific hybrids such as ‘Vidal blanc’ in the lower midwestern United States as well as in other regions with long, warm growing seasons. Balanced pruning to 30 nodes per 454 g of dormant prunings and cluster thinning to two clusters per shoots optimized yield, maintained fruit composition, improved primary bud cold hardiness, and achieved an optimum fruit weight-to-pruning weight ratio of 10.0 kg·kg–1. Thus, this approach should be used for ’Vidal blanc’ in the lower midwestern United States to sustain production.