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AlexNet
View on Wikipedia| AlexNet | |
|---|---|
| Developers | Alex Krizhevsky, Ilya Sutskever, and Geoffrey Hinton |
| Initial release | June 28, 2011 |
| Repository | code |
| Written in | CUDA, C++ |
| Type | Convolutional neural network |
| License | New BSD License |


AlexNet is a convolutional neural network architecture developed for image classification tasks, notably achieving prominence through its performance in the ImageNet Large Scale Visual Recognition Challenge (ILSVRC). It classifies images into 1,000 distinct object categories and is regarded as the first widely recognized application of deep convolutional networks in large-scale visual recognition.
Developed in 2012 by Alex Krizhevsky in collaboration with Ilya Sutskever and his Ph.D. advisor Geoffrey Hinton at the University of Toronto, the model contains 60 million parameters and 650,000 neurons.[1] The original paper's primary result was that the depth of the model was essential for its high performance, which was computationally expensive, but made feasible due to the utilization of graphics processing units (GPUs) during training.[1]
The three formed team SuperVision and submitted AlexNet in the ImageNet Large Scale Visual Recognition Challenge on September 30, 2012.[2] The network achieved a top-5 error rate of 15.3% to win the contest, more than 10.8% above the runner-up.
The architecture influenced a large number of subsequent work in deep learning, especially in applying neural networks to computer vision.
Architecture
[edit]AlexNet contains eight layers: the first five are convolutional layers, some of them followed by max-pooling layers, and the last three are fully connected layers. The network, except the last layer, is split into two copies, each run on one GPU, because the network did not fit the VRAM of a single Nvidia GTX 580 3GB GPU.[1]: Section 3.2 The entire structure can be written as
(CONV → RN → MP)2 → (CONV3 → MP) → (FC → DO)2 → Linear → softmax
where
- CONV = convolutional layer (with ReLU activation)
- RN = local response normalization
- MP = max-pooling
- FC = fully connected layer (with ReLU activation)
- Linear = fully connected layer (without activation)
- DO = dropout
Notably, the convolutional layers 3, 4, 5 were connected to one another without any pooling or normalization. It used the non-saturating ReLU activation function, which trained better than tanh and sigmoid.[1]
Training
[edit]The ImageNet training set contained 1.2 million images. The model was trained for 90 epochs over a period of five to six days using two Nvidia GTX 580 GPUs (3GB each).[1] These GPUs have a theoretical performance of 1.581 TFLOPS in float32 and were priced at US$500 upon release.[3] Each forward pass of AlexNet required approximately 1.43 GFLOPs.[4] Based on these values, the two GPUs together were theoretically capable of performing over 2,200 forward passes per second under ideal conditions.
The dataset images were stored in JPEG format. They took up 27GB of disk. The neural network took up 2GB of RAM on each GPU, and around 5GB of system RAM during training. The GPUs were responsible for training, while the CPUs were responsible for loading images from disk, and data-augmenting the images.[5]
AlexNet was trained with momentum gradient descent with a batch size of 128 examples, momentum of 0.9, and weight decay of 0.0005. Learning rate started at 10−2 and was manually decreased 10-fold whenever validation error appeared to stop decreasing. It was reduced three times during training, ending at 10−5.
It used two forms of data augmentation, both computed on the fly on the CPU, thus "computationally free":
- Each image from ImageNet was first scaled, so that its shorter side was of length 256. Then the central 256×256 patch was cropped out and normalized (dividing the pixel values so that they fall between 0 and 1, then subtracting by [0.485, 0.456, 0.406], then dividing by [0.229, 0.224, 0.225]. These are the mean and standard deviations for ImageNet, so this whitens the input data).
- Extracting random 224×224 patches (and their horizontal reflections) from the 256×256 crop. This increases the size of the training set 2048-fold.
- Randomly shifting the RGB value of each image along the three principal directions of the RGB values of its pixels.
The resolution 224×224 was picked, because 256 - 16 - 16 = 224, meaning that given a 256×256 image, framing out a width of 16 on its 4 sides results in a 224×224 image.
It used local response normalization, and dropout regularization with drop probability 0.5.
All weights were initialized as gaussians with 0 mean and 0.01 standard deviation. Biases in convolutional layers 2, 4, 5, and all fully-connected layers, were initialized to constant 1 to avoid the dying ReLU problem.
At test time, to use a trained AlexNet for predicting the class of an image, that image is first scaled, so that its shorter side was of length 256. Then the central 256×256 patch was cropped out. Then, the five 224 × 224 patches (the four corner patches and the center patch) as well as their horizontal reflections are computed, 10 patches in all. The network's predicted probabilities on all 10 patches are averaged, and that is the final predicted probability.
ImageNet competition
[edit]The version they used to enter the 2012 ImageNet competition was an ensemble of 7 AlexNets.
Specifically, they trained 5 AlexNets of the previously described architecture (with 5 CONV layers) on the ILSVRC-2012 training set (1.2 million images). They also trained 2 variant AlexNets, obtained by adding one extra CONV layer over the last pooling layer. These were trained by first training on the entire ImageNet Fall 2011 release (15 million images in 22K categories), and then finetuning it on the ILSVRC-2012 training set. The final system of 7 AlexNets was used by averaging their predicted probabilities.
History
[edit]Previous work
[edit]
(AlexNet image size should be 227×227×3, instead of 224×224×3, so the math will come out right. The original paper said different numbers, but Andrej Karpathy, the former head of computer vision at Tesla, said it should be 227×227×3 (he said Alex didn't describe why he put 224×224×3). The next convolution should be 11×11 with stride 4: 55×55×96 (instead of 54×54×96). It would be calculated, for example, as: [(input width 227 - kernel width 11) / stride 4] + 1 = [(227 - 11) / 4] + 1 = 55. Since the kernel output is the same length as width, its area is 55×55.)
In 1980, Kunihiko Fukushima proposed an early CNN named neocognitron.[6][7] It was trained by an unsupervised learning algorithm. The LeNet-5 (Yann LeCun et al., 1989)[8][9] was trained by supervised learning with backpropagation algorithm, with an architecture that is essentially the same as AlexNet on a small scale.
Max pooling was used in 1990 for speech processing (essentially a 1-dimensional CNN),[10] and for image processing, was first used in the Cresceptron of 1992.[11]
During the 2000s, as GPU hardware improved, some researchers adapted these for general-purpose computing, including neural network training. (K. Chellapilla et al., 2006) trained a CNN on GPU that was 4 times faster than an equivalent CPU implementation.[12] (Raina et al 2009) trained a deep belief network with 100 million parameters on an Nvidia GeForce GTX 280 at up to 70 times speedup over CPUs.[13] A deep CNN of (Dan Cireșan et al., 2011) at IDSIA was 60 times faster than an equivalent CPU implementation.[14] Between May 15, 2011, and September 10, 2012, their CNN won four image competitions and achieved state of the art for multiple image databases.[15][16][17] According to the AlexNet paper,[1] Cireșan's earlier net is "somewhat similar". Both were written with CUDA to run on GPU.
Computer vision
[edit]During the 1990–2010 period, neural networks were not better than other machine learning methods like kernel regression, support vector machines, AdaBoost, structured estimation,[18] among others. For computer vision in particular, much progress came from manual feature engineering, such as SIFT features, SURF features, HoG features, bags of visual words, etc. It was a minority position in computer vision that features can be learned directly from data, a position which became dominant after AlexNet.[19]
In 2011, Geoffrey Hinton started reaching out to colleagues about "What do I have to do to convince you that neural networks are the future?", and Jitendra Malik, a sceptic of neural networks, recommended the PASCAL Visual Object Classes challenge. Hinton said its dataset was too small, so Malik recommended to him the ImageNet challenge.[20]
The ImageNet dataset, which became central to AlexNet's success, was created by Fei-Fei Li and her collaborators beginning in 2007. Aiming to advance visual recognition through large-scale data, Li built a dataset far larger than earlier efforts, ultimately containing over 14 million labeled images across 22,000 categories. The images were labeled using Amazon Mechanical Turk and organized via the WordNet hierarchy. Initially met with skepticism, ImageNet later became the foundation of the ImageNet Large Scale Visual Recognition Challenge (ILSVRC) and a key resource in the rise of deep learning.[21]
Sutskever and Krizhevsky were both graduate students. Before 2011, Krizhevsky had already written cuda-convnet to train small CNNs on CIFAR-10 with a single GPU. Sutskever convinced Krizhevsky, who could do GPGPU well, to train a CNN on ImageNet, with Hinton serving as principal investigator. So Krizhevsky extended cuda-convnet for multi-GPU training. AlexNet was trained on 2 Nvidia GTX 580 in Krizhevsky's bedroom at his parents' house. During 2012, Krizhevsky performed hyperparameter optimization on the network until it won the ImageNet competition later the same year. Hinton commented that, "Ilya thought we should do it, Alex made it work, and I got the Nobel Prize".[22] At the 2012 European Conference on Computer Vision, following AlexNet's win, researcher Yann LeCun described the model as "an unequivocal turning point in the history of computer vision".[21]
AlexNet's success in 2012 was enabled by the convergence of three developments that had matured over the previous decade: large-scale labeled datasets, general-purpose GPU computing, and improved training methods for deep neural networks. The availability of ImageNet provided the data necessary for training deep models on a broad range of object categories. Advances in GPU programming through Nvidia's CUDA platform enabled practical training of large models. Together with algorithmic improvements, these factors enabled AlexNet to achieve high performance on large-scale visual recognition benchmarks.[21] Reflecting on its significance over a decade later, Fei-Fei Li stated in a 2024 interview: "That moment was pretty symbolic to the world of AI because three fundamental elements of modern AI converged for the first time".[21]
While AlexNet and LeNet share essentially the same design and algorithm, AlexNet is much larger than LeNet and was trained on a much larger dataset on much faster hardware. Over the period of 20 years, both data and compute became cheaply available.[19]
Subsequent work
[edit]AlexNet is highly influential, resulting in much subsequent work in using CNNs for computer vision and using GPUs to accelerate deep learning. As of early 2025, the AlexNet paper has been cited over 184,000 times according to Google Scholar.[23]
At the time of publication, there was no framework available for GPU-based neural network training and inference. The codebase for AlexNet was released under a BSD license, and had been commonly used in neural network research for several subsequent years.[24][19]
In one direction, subsequent works aimed to train increasingly deep CNNs that achieve increasingly higher performance on ImageNet. In this line of research are GoogLeNet (2014), VGGNet (2014), Highway network (2015), and ResNet (2015). Another direction aimed to reproduce the performance of AlexNet at a lower cost. In this line of research are SqueezeNet (2016), MobileNet (2017), EfficientNet (2019).
Geoffrey Hinton, Ilya Sutskever, and Alex Krizhevsky formed DNNResearch soon afterwards and sold the company, and the AlexNet source code along with it, to Google. There had been improvements and reimplementations for the AlexNet, but the original version as of 2012, at the time of its winning of ImageNet, had been released under BSD-2 license via Computer History Museum.[25]
References
[edit]- ^ a b c d e f Krizhevsky, Alex; Sutskever, Ilya; Hinton, Geoffrey E. (2017-05-24). "ImageNet classification with deep convolutional neural networks" (PDF). Communications of the ACM. 60 (6): 84–90. doi:10.1145/3065386. ISSN 0001-0782. S2CID 195908774.
- ^ "ImageNet Large Scale Visual Recognition Competition 2012 (ILSVRC2012)". image-net.org.
- ^ "NVIDIA GeForce GTX 580 Specs". TechPowerUp. 2024-11-12. Retrieved 2024-11-12.
- ^ "calflops: a FLOPs and Params calculate tool for neural networks". pypi.org. Retrieved 2024-12-10.
- ^ SuperVision team invited talk at the ImageNet Large Scale Visual Recognition Challenge 2012 (ILSVRC2012) workshop on 2012-10-12, held within the ECCV 2012, in Florence, Italy.
- ^ Fukushima, K. (2007). "Neocognitron". Scholarpedia. 2 (1): 1717. Bibcode:2007SchpJ...2.1717F. doi:10.4249/scholarpedia.1717.
- ^ Fukushima, Kunihiko (1980). "Neocognitron: A Self-organizing Neural Network Model for a Mechanism of Pattern Recognition Unaffected by Shift in Position" (PDF). Biological Cybernetics. 36 (4): 193–202. doi:10.1007/BF00344251. PMID 7370364. S2CID 206775608. Retrieved 16 November 2013.
- ^ LeCun, Y.; Boser, B.; Denker, J. S.; Henderson, D.; Howard, R. E.; Hubbard, W.; Jackel, L. D. (1989). "Backpropagation Applied to Handwritten Zip Code Recognition" (PDF). Neural Computation. 1 (4). MIT Press - Journals: 541–551. doi:10.1162/neco.1989.1.4.541. ISSN 0899-7667. OCLC 364746139.
- ^ LeCun, Yann; Léon Bottou; Yoshua Bengio; Patrick Haffner (1998). "Gradient-based learning applied to document recognition" (PDF). Proceedings of the IEEE. 86 (11): 2278–2324. CiteSeerX 10.1.1.32.9552. doi:10.1109/5.726791. S2CID 14542261. Retrieved October 7, 2016.
- ^ Yamaguchi, Kouichi; Sakamoto, Kenji; Akabane, Toshio; Fujimoto, Yoshiji (November 1990). A Neural Network for Speaker-Independent Isolated Word Recognition. First International Conference on Spoken Language Processing (ICSLP 90). Kobe, Japan. Archived from the original on 2021-03-07. Retrieved 2019-09-04.
- ^ Weng, J.; Ahuja, N.; Huang, T.S. (1992). Cresceptron: a self-organizing neural network which grows adaptively (PDF). Vol. 1. IEEE. pp. 576–581. doi:10.1109/IJCNN.1992.287150. ISBN 978-0-7803-0559-5.
- ^ Kumar Chellapilla; Sidd Puri; Patrice Simard (2006). "High Performance Convolutional Neural Networks for Document Processing". In Lorette, Guy (ed.). Tenth International Workshop on Frontiers in Handwriting Recognition. Suvisoft.
- ^ Raina, Rajat; Madhavan, Anand; Ng, Andrew Y. (2009-06-14). Large-scale deep unsupervised learning using graphics processors. ACM. pp. 873–880. doi:10.1145/1553374.1553486. ISBN 978-1-60558-516-1.
- ^ Cireșan, Dan; Ueli Meier; Jonathan Masci; Luca M. Gambardella; Jurgen Schmidhuber (2011). "Flexible, High Performance Convolutional Neural Networks for Image Classification" (PDF). Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence-Volume Volume Two. 2: 1237–1242. Retrieved 17 November 2013.
- ^ "IJCNN 2011 Competition result table". OFFICIAL IJCNN2011 COMPETITION. 2010. Retrieved 2019-01-14.
- ^ Schmidhuber, Jürgen (17 March 2017). "History of computer vision contests won by deep CNNs on GPU". Retrieved 14 January 2019.
- ^ Cireșan, Dan; Meier, Ueli; Schmidhuber, Jürgen (June 2012). "Multi-column deep neural networks for image classification". 2012 IEEE Conference on Computer Vision and Pattern Recognition. New York, NY: Institute of Electrical and Electronics Engineers (IEEE). pp. 3642–3649. arXiv:1202.2745. CiteSeerX 10.1.1.300.3283. doi:10.1109/CVPR.2012.6248110. ISBN 978-1-4673-1226-4. OCLC 812295155. S2CID 2161592.
- ^ Taskar, Ben; Guestrin, Carlos; Koller, Daphne (2003). "Max-Margin Markov Networks". Advances in Neural Information Processing Systems. 16. MIT Press.
- ^ a b c Zhang, Aston; Lipton, Zachary; Li, Mu; Smola, Alexander J. (2024). "8.1. Deep Convolutional Neural Networks (AlexNet)". Dive into deep learning. Cambridge New York Port Melbourne New Delhi Singapore: Cambridge University Press. ISBN 978-1-009-38943-3.
- ^ Li, Fei Fei (2023). The worlds I see: curiosity, exploration, and discovery at the dawn of AI (First ed.). New York: Moment of Lift Books ; Flatiron Books. ISBN 978-1-250-89793-0.
- ^ a b c d "How a stubborn computer scientist accidentally launched the deep learning boom". Ars Technica. 11 November 2024. Retrieved 24 March 2025.
- ^ hhackford (2025-03-20). "CHM Releases AlexNet Source Code". CHM. Retrieved 2025-03-22.
- ^ AlexNet paper on Google Scholar
- ^ Krizhevsky, Alex (July 18, 2014). "cuda-convnet: High-performance C++/CUDA implementation of convolutional neural networks". Google Code Archive. Retrieved 2024-10-20.
- ^ computerhistory/AlexNet-Source-Code, Computer History Museum, 2025-03-22, retrieved 2025-03-22
AlexNet
View on GrokipediaBackground
Historical Context in Computer Vision
Early computer vision research relied heavily on hand-crafted features to represent images, as these methods aimed to capture invariant properties like edges, textures, and shapes manually designed by researchers. Techniques such as Scale-Invariant Feature Transform (SIFT), introduced in 2004, detected and described local features robust to scale and rotation changes, enabling tasks like object recognition and image matching.[4] Similarly, Histograms of Oriented Gradients (HOG), proposed in 2005, focused on gradient orientations to detect objects like pedestrians by emphasizing edge directions in localized portions of an image.[5] These features were typically fed into shallow machine learning models, such as support vector machines (SVMs), which performed classification based on predefined descriptors rather than learning hierarchical representations from raw pixels.[6] In the 2000s, these approaches faced significant challenges due to the high-dimensional nature of image data, where the "curse of dimensionality" led to sparse representations and difficulties in capturing complex semantic information. Hand-crafted features often struggled with variability in lighting, viewpoint, and occlusion, requiring extensive engineering to generalize across diverse scenarios, while shallow classifiers like SVMs were prone to overfitting on large datasets with millions of pixels.[6] Traditional methods also exhibited limited scalability, as manual feature design became increasingly labor-intensive for real-world applications involving natural images, hindering progress in tasks like large-scale object detection.[7] Neural networks, revitalized by the backpropagation algorithm in 1986, offered a promising alternative for learning features automatically but entered a period of dormancy in the 1990s amid the broader "AI winter," primarily due to insufficient computational power for training deep architectures on complex data.[8][9] Limited hardware constrained networks to small scales, such as Yann LeCun's LeNet in 1998, a convolutional neural network designed for handwritten digit recognition on low-resolution grayscale images like those in the MNIST dataset. This milestone demonstrated gradient-based learning for simple pattern recognition but highlighted the era's constraints, as deeper networks remained impractical without advances in processing capabilities. The emergence of large-scale challenges like the ImageNet competition in 2010 served as a catalyst for renewed interest in scalable deep learning solutions.[10]ImageNet Dataset and Competition
The ImageNet project was initiated in 2009 by Fei-Fei Li and her collaborators at Stanford University and Princeton University to address the lack of large-scale, annotated image datasets for computer vision research.[11] Drawing from the WordNet lexical database, ImageNet organizes images hierarchically into synsets representing concepts, primarily nouns, with the goal of populating over 80,000 categories.[11] By its completion, the dataset encompassed over 14 million annotated images across approximately 21,841 categories, crowdsourced via Amazon Mechanical Turk for labeling to ensure scalability and diversity.[12] This vast repository enabled researchers to train models on realistic, varied visual data, far exceeding prior datasets like Caltech-101 or PASCAL VOC in size and complexity.[11] To foster advancements in visual recognition, the ImageNet Large Scale Visual Recognition Challenge (ILSVRC) was launched in 2010 as an annual competition hosted alongside the PASCAL VOC workshop. The challenge utilized a curated subset of ImageNet, known as ILSVRC2010 data, comprising 1,000 categories (WNIDs from the WordNet hierarchy) with about 1.2 million training images, 50,000 validation images, and 100,000 test images sourced from Flickr and other engines, all hand-annotated for object presence. The primary metric was the top-5 error rate, where a prediction succeeds if the correct class is among the five highest-ranked outputs, emphasizing practical recognition performance over exact top-1 accuracy. This setup standardized evaluation, allowing direct comparison of algorithms on a massive scale and motivating innovations in feature extraction and classification.[13] In the inaugural 2010 and 2011 ILSVRC editions, winning approaches relied on shallow, hand-engineered methods rather than deep learning, underscoring the computational and methodological limitations of the era. For instance, the 2010 victor employed linear support vector machines (SVMs) trained on SIFT and LBP features, yielding a top-5 error rate of 28.1%, while the 2011 winner combined compressed Fisher vectors with SVMs for a 25.7% error rate. These techniques, which processed images via local feature detectors like SIFT or HOG followed by bag-of-words encoding and shallow classifiers, highlighted the need for end-to-end learning systems capable of handling the dataset's scale without manual feature design. The 2012 ILSVRC edition expanded to include two parallel tracks—image classification (focusing on category labeling) and classification with localization (requiring bounding box predictions for objects)—to evaluate both recognition and spatial understanding.[14] Participation grew significantly from prior years, drawing teams from academia and industry, with the event offering cash prizes sponsored by tech companies like Google and Microsoft to incentivize high-quality submissions. This structure not only tested algorithmic robustness on the 1,000-class subset but also amplified ImageNet's role as a benchmark, spurring scalable deep learning solutions amid increasing computational resources.[13]Architecture
Overall Design
AlexNet is a deep convolutional neural network (CNN) designed for large-scale image classification, comprising eight layers in total: five convolutional layers and three fully connected layers.[1] The network accepts input images of size 224 × 224 pixels with three color channels (RGB), which are preprocessed by cropping and resizing from larger originals to fit this resolution.[1] It processes these inputs through the layers to produce output probabilities over 1,000 classes corresponding to the ImageNet challenge categories, achieved via a final softmax layer.[1] The layer sequence begins with convolutional layers (Conv1 through Conv5) for hierarchical feature extraction, interspersed with max-pooling operations after Conv1, Conv2, and Conv5 to provide spatial invariance and dimensionality reduction.[1] Following the convolutional and pooling stages, the feature maps are flattened and fed into three fully connected layers (FC6, FC7, and FC8), where FC8 connects to the output softmax.[1] This structure progressively reduces the spatial dimensions from the initial 224 × 224 to 6 × 6 feature maps before the fully connected layers, primarily through strided convolutions and max-pooling with kernel size 3 and stride 2.[1] In terms of scale, AlexNet contains approximately 60 million parameters and around 650,000 neurons, with the majority of parameters concentrated in the fully connected layers due to their dense connectivity.[1] During the forward pass, convolutional layers apply learnable filters to detect local patterns such as edges and textures, building increasingly complex representations across depths, while max-pooling summarizes these features to promote translation invariance.[1] ReLU (Rectified Linear Unit) activations are applied after each convolutional and fully connected layer (except the output softmax) to introduce nonlinearity and accelerate convergence.[1]Key Innovations
One of the primary innovations in AlexNet was the adoption of rectified linear units (ReLUs) as the activation function throughout the network, replacing traditional sigmoid or hyperbolic tangent functions. ReLUs, defined as , enable faster training convergence—approximately six times faster than tanh units in similar models—and mitigate the vanishing gradient problem by allowing gradients to flow more effectively through the network during backpropagation. This choice was inspired by prior work demonstrating ReLUs' benefits in deep architectures, and it contributed significantly to AlexNet's ability to train a deep network without getting trapped in poor local minima.[15] To handle the computational demands of the large model, AlexNet employed GPU parallelization by training on two NVIDIA GTX 580 GPUs, each with 3 GB of memory. The network was parallelized by splitting the kernels across the two GPUs (half on each), with connections in layers 2, 4, and 5 limited to the same GPU's previous layer kernels, and full connections in layer 3; the GPUs communicated only at layer boundaries to pass activations, enabling efficient processing without inter-GPU synchronization during forward and backward passes. This setup reduced training time to five or six days, making deep learning feasible on consumer-grade hardware at the time and demonstrating the scalability of convolutional neural networks through hardware acceleration.[15] Overfitting was addressed through dropout regularization applied to the two largest fully connected layers, where individual neurons were randomly inactivated during training with a probability of 0.5, effectively preventing co-adaptation of features and simulating an ensemble of thinner networks. This technique, integrated without other regularization methods, substantially improved generalization on the ImageNet dataset. Complementing this, data augmentation expanded the effective training set size by a factor of over 2000: random 224×224 crops were extracted from 256×256 images (including horizontal flips with 50% probability), and color jittering was applied via principal component analysis (PCA) on the RGB channels, adding variations with eigenvalues capturing 90% of the variance to enhance robustness to lighting and color shifts.[15] Additionally, local response normalization (LRN) was introduced after the first and second convolutional layers to promote sparsity and competitive inhibition among neighboring feature maps, drawing from biological vision systems. For a neuron with activity in a local neighborhood of size , the normalized response is given bywith parameters , , and , where the sum is over adjacent channels at the same spatial location; this normalization helped improve performance by about 1.2% on the validation set compared to models without it.[15]