text:
\section{Introduction}
Deep Learning Networks (DLNs) have exhibited better than human performance in several vision-related tasks \cite{lecun2015deep}. However, they have been recently shown to be vulnerable toward adversarial attacks \cite{kurakin2016adversarial, goodfellowexplaining, panda2018explainable}: slight changes of input pixel intensities that fool a DLN to misclassify an input with high confidence (Fig. 1). What is more worrying is that such small changes (that craft adversaries) are visually imperceptible to humans, yet, mislead a DLN. This vulnerability severely limits the potential safe-use and deployment of DLNs in real-world scenarios. For instance, an attacker can fool a DLN deployed on a self-driving car to mispredict a STOP sign as a GO signal, and cause fatal accidents.

Subsequently, there have been several theories pertaining to the adversarial susceptibility of DLNs \cite{goodfellowexplaining}. The most common one suggests that the presence of adversary is an outcome of the excessive linearity of a DLN (a property of high-dimensional dot-products). While one can argue that ReLU-type activation imposes non-linearity in a model, the linear operations such as Convolution, Pooling exceed the number of non-linear ReLU operations. Further, ReLU is typically a linear functionality in the $>0$ regime, and hence, plagues a DLN to be sufficiently linear. Now, this linearity causes a model to extrapolate its behavior for points in the hyper-space (of data and model parameters) that lie outside the training/test data manifold. Adversarial inputs, essentially, are images that are synthesized such that they 'lie far' from the typical data manifold and hence get misclassified.
\setlength{\textfloatsep}{6pt plus 1.0pt minus 8.0pt}
\begin{figure}
\centering
\includegraphics[width=0.7\linewidth]{figures/fig1}
\caption{\textit{An image of a \textbf{ship} perturbed with adversarial noise yields an adversarial image that fools the classifier. The classifier predicts the original image correctly with a confidence of 92\%, while gets fooled by the adversarial image mispredicting it as \textbf{plane} with a high confidence of 91\%.}}
\end{figure}

Fig. 2 (a) demonstrates this data manifold intuition and adversarial input creation with a cartoon. Since DLNs are discriminative models, they partition a very high-dimensional input space into different classes by learning appropriate decision boundaries. The class-specific decision boundaries simply divide the space into hyper-volumes. Interestingly, these hyper-volumes encompass the training data examples as well as large areas of unpopulated space that is arbitrary and untrained. This extrapolation of decision boundary beyond the training data space is a result of \textit{'linearity'}, that in turn, gives rise to generalization ability. The fact that a model trained only on training data is able to predict well on unseen test data (termed as, generalization) is a favorable outcome of this extrapolation property. Unfortunately, this property also exposes a model to adversarial attacks. Adversarial data are created by simply adding small perturbations to an input data point, that shifts it from its manifold (or hyper-volume) to a different hyper-volume (that the model has not been trained upon and shows extrapolated behavior), causing misclassification.

From the above intuition, one can deduce that adding regularization features to a DLN's training will improve its generalization ability and in turn, adversarial robustness. In fact, the most effective form of adversarial defense so far is training a model with adversarial data augmentation (called adversarial training) \cite{szegedy2013intriguing}. It is evident that explicit training on adversarial data will increase the model's capability to generalize and hence predict correctly on unseen adversarial data. However, the above discussion on excessive linearity and hyper-space dimensionality points to an alternate and unexplored regularization possibility, that is discretizing or constraining the data manifold for achieving adversarial robustness. For instance, discretizing the input data (say from 256-pixel value levels (or 8-bit) to 4 levels (or 2-bit)) reduces the regions into which data can be perturbed. In other words, the minimum perturbation required to shift a particular data point from one hyper-volume to another will increase in a discretized space (Fig. 2 (b)). This in turn will intrinsically improve the resistance of a DLN. Similarly, discretizing the parameter space (as in binarized neural networks (BNNs) \cite{hubara2016binarized}) will introduce discontinuities and quantization in the manifold (that is non-linear by nature). This will further decrease the extent of hyper-volume space that is arbitrary/untrained and thus reduce adversarial susceptibility (Fig. 2(b)). It is evident that such discretization methods have an added advantage of computational efficiency. In fact, low-precision neural networks (BNNs and related XNOR-Nets \cite{rastegari2016xnor}) were introduced with a key motif of obtaining reduced memory and power-consumption for hardware deployment of DLNs.
\setlength{\textfloatsep}{5pt plus 1.0pt minus 4.0pt}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{figures/fig2}
\caption{\textit{Cartoon of the intuition behind adversary creation and discretization. (a) The data points (shown as 'dots') encompass the data manifold in the high-dimensional subspace. The classifier is trained to separate the data into different categories or hyper-volumes based on which the decision boundary is formed. Note, the decision boundary is a characteristic of the trained parameters (weights) of the model. The decision boundary is, however, extrapolated to vast regions of the high-dimensional subspace that are unpopulated and untrained because of \textbf{linear} model behavior. Adversaries are created by perturbing the data points into these empty regions or hyper-volumes and are thus misclassified (\textbf{orange} mispredicted as \textbf{green} in this case). (b) Discretization quantizes the data manifold thereby introducing a minimum perturbation required to shift a data point. As quantization will increase, so will the minimum allowed distortion. Further, discretization constrains the creation of adversaries since not all transitions can cause a data point to shift between hyper-volumes.}}
\end{figure}

In this paper, we demonstrate that discretization, besides offering obvious efficiency improvements, has far-reaching implication on a model's adversarial resistance. We particularly emphasize on three different discretization themes and illustrate their suitability toward improving a DLN's adversarial robustness, as follows:

\begin{itemize}
\item{\textit{Discretization of input space:} We reduce the input dimensionality by quantizing the RGB pixel intensities into a variable range: $2^8 =256$ to $2^2=4$. We show that for minimal loss in accuracy, the adversarial robustness of a model substantially improves ($<1\%$ accuracy difference between clean test and adversarial test data), even, without any adversarial training. Furthermore, we show that combining adversarial training with 2-bit input discretization makes a model substantially more robust (than adversarial training with full 8-bit input precision) for large perturbation ranges.}
\item{\textit{Discretization of parameter space:} We show that models trained with low-precision weights and activations, such as BNNs, are intrinsically more robust to adversarial perturbations than full precision networks. Furthermore, we find that training BNNs with adversarial data augmentation is difficult. However, increasing the capacity of the BNN (with more neurons and weights) minimizes the adversarial training difficulty. For sufficient model capacity, adversarially trained BNNs yield higher adversarial robustness than their full-precision counterpart.}
\item{\textit{Discretization of both input \& parameter space:} We demonstrate that combining input discretization with binarized weight /activation training greatly improves a model's robustness. In fact, training a BNN with input discretization (say, 2-bit input) yields similar or better adversarial accuracy as that of an adversarially trained full-precision model. Thus, the combined discretization scheme can be seen as an efficient alternative to achieving adversarial robustness without the expensive data augmentation procedure.}
\end{itemize}

\section{Related Work}
Based on the intuition demonstrated in Fig. 2, robustness of DLNs can be attributed to two factors: property of the input and model property. Consequently, there have been many recent proposals \cite{guo2018countering, buckman2018thermometer, chen2018improving,  xu2017feature} that exploit the input-dependent factor and try to remove adversarial perturbations by applying input preprocessing or transformations. Due to the simplicity of this approach, these methods are attractive for practical implementations as they do not incur large computational overhead (as with adversarial training) and do not interfere with the learning process. Our paper complements the results of the prior works while presenting a novel result on the effectiveness of combined parameter and input discretization for adversarial robustness.

One of the first works on input discretization by Xu et al. \cite{xu2017feature} propose a \textit{depth-color-squeezing} technique wherein they reduce the degrees of freedom available to an adversary by `removing' unnecessary features. Our pixel discretization scheme is based on their color depth reduction technique. However, the key idea in \cite{xu2017feature} is to compare the model's prediction on the original input with its prediction on the squeezed input during testing. Xu et al. train the model with regular inputs and during inference use pixel discretization to detect adversarial inputs. That is, if the original and squeezed inputs produce predictions with large difference (greater than an user-defined threshold), the model deems the input to be adversarial and rejects it. Ultimately, the model outputs prediction for only legitimate or non-adversarial inputs. In contrast, the key novel aspect of our work is to train a model with discretized pixel data such that the model looks at a reduced input subspace during training that decreases or constrains its' ability to overly genralize in the high-dimensional subspace. Similarly, the thermometer encoding technique and input transformation technique proposed in \cite{buckman2018thermometer}, \cite{guo2018countering} are guided by the same intuition of reducing adversarial subspace dimensions. Guo et al. \cite{guo2018countering} trained the network with images transformed in various ways and observed improved adversarial resistance. However, they measured robustness for controlled graybox attack settings (where, the model parameters are known to the attacker but the input transformations are unknown). Our paper's results on whitebox attack is a stronger notion of robustness as we assume all parameters as well as input discretization known to the attacker. Thus, our results while supporting the claims of \cite{guo2018countering} are more substantial and generalizable. In Buckman et al. \cite{buckman2018thermometer}, the authors propose a thermometer encoding technique to map input pixels to a binary vector in order to make more meaningful change during pixel discretization without losing any information from the original image. While, the authors show good adversarial robustness results on small tasks, such as, MNIST \cite{lecun1998mnist}, they are shown to achieve poor performance on more complex datasets (like, CIFAR10 \cite{krizhevsky2009cifar}) \cite{chen2018improving}.

In order to address the limitation of the prior works solely based on input space modifications, we investigate the effect of combining model discretization with the input transformation property thereby leveraging both criteria that contribute to adversarial dimensions. To the best of our knowledge, we are the first to formally evaluate and analyze the impact of input and parameter space discretization for DLNs (across simple and complex datasets including ImageNet2012 dataset) on robustness. A recent work \cite{galloway2018attacking} demonstrated the effectiveness of BNNs against adversarial attacks and observed a similar difficulty in adversarial training with BNNs. However, they did not consider input space discretization and its impact on robustness. While complementing their results, we show that quantizing the input pixels of a BNN during training greatly improves its robustness, even waiving the need for the expensive and time-consuming adversarial training, for certain perturbation ranges.

\section{Background on Adversarial \\Attacks}
\textit{Generating Adversaries :} Adversarial examples are created using a trained DLN's parameters and gradients. As shown in Fig. 1, the adversarial perturbation, $\Delta$, is not just some random noise, but carefully designed to bias the network's prediction on a given input towards a wrong class. Goodfellow et. al \cite{goodfellowexplaining} proposed a simple method called Fast Gradient Sign Method (FGSM) to craft adversarial examples by linearizing a trained model's loss function ($\mathcal L$, say cross-entropy) with respect to the input ($X$):
\begin{equation}
X_{adv} = X + \epsilon \times sign(\nabla_X \mathcal L(\theta, X ,y_{true}))
\end{equation}
Here, $y_{true}$ is the true class label for the input $X$, $\theta$ denotes the model parameters (weights, biases etc.) and $\epsilon$ quantifies the magnitude of distortion. The net perturbation added to the input ($\Delta = \epsilon \times sign(\nabla_X \mathcal L(\theta, X ,y_{true}))$) is, thus, regulated by $\epsilon$. Distorting the input image in the direction of steepest gradient has the maximal effect on the loss function during prediction. Intuitively, referring to Fig.2, this distortion shifts the data point from the trained region or hyper-volume to an arbitrary region thereby fooling the model.

\textit{Types of Attacks :} In machine learning literature, there are two kinds of attacks: Black-Box (BB), White-Box (WB) that are used to study adversarial robustness \cite{tramer2018ensemble}. WB adversaries are created using the \textit{target} model's parameters, that is, the attacker has full knowledge of a target model's training information.  BB attacks refer to the case when the attacker has no knowledge about the target model's parameters. In this case, adversaries are created using a different \textit {source} model's parameters trained on the same classification task as the target model. Since BB attacks are transferred onto the target model, they are weaker than WB attacks. Security against WB attacks is a stronger notion and robustness against WB attacks guarantees robustness against BB for similar perturbation ($\epsilon$) range.

\textit{Adversarial Training :} Adversarial training simply injects adversarial examples into the training dataset of a model \cite{szegedy2013intriguing}. For each training sample in the dataset, an adversary is created using FGSM \cite{goodfellowexplaining}. There are several forms of adversarial training. For instance, instead of using the same $\epsilon$ for all training examples, \cite{kurakin2016adversariala, tramer2018ensemble} propose to sample a unique $\epsilon$ (from a random normal distribution) for each training example. This will increase the variation in the adversaries created thereby increasing the robustness of a network to larger range of $\epsilon$ values. The authors in \cite{madry2018towards} use WB adversaries created, using a multi-step variant of FGSM to guarantee a strong defense against both BB and WB attacks. Note, the common theme across all adversarial training methods is data augmentation.

In this work, we focus on adversarial attacks created using FGSM and evaluate the robustness of models against WB adversaries. We evaluate a model's robustness/report adversarial accuracy on the adversarial dataset created using the test data for a given task.

\section{Experiments}
We conduct a series of experiments for each discretization theme, primarily using MNIST \cite{lecun1998mnist} (Fully Connected Network, FCN) and CIFAR10 \cite{krizhevsky2009cifar} (AlexNet \cite{gitbnn} architecture), detailing the advantages and limitation of each approach. We compare the adversarial robustness of each discretization approach with its' full-precision counterpart (with and without adversarial training), using $\epsilon$ values reported in recent works \cite{tramer2018ensemble,madry2018towards}. For adversarial training, we employ Random-step FGSM (R-FGSM) proposed in \cite{tramer2018ensemble} to create a variety of training set adversaries. R-FGSM perturbs the input $X$ with a small random step (sampled from a normal distribution $\mathcal N$) before adding the loss gradient to the input: $X_{adv} = X' + \Delta$, where $X' = X + \alpha sign(\mathcal N(0, I), \alpha = \epsilon/2$. We use WB adversarial training to confer strong robustness toward all forms of attacks. Note, for MNIST (CIFAR), we use $\epsilon = 0.3$ $(8/255)$ during adversarial training. For evaluating the robustness of parameter space discretization, we use BNNs \cite{hubara2016binarized, gitbnn} to evaluate CIFAR10 and MNIST datasets. We also evaluate the robustness of discretization methods on large-scale datasets, CIFAR100 (ResNet20 architecture \cite{he2016deep}) and Imagenet \cite{deng2009imagenet} (AlexNet architecture) using XNOR networks \cite{rastegari2016xnor, gitxnor}. Please note, for MNIST we use two different FCN architectures: FCN1-4 hidden layer network with 6144 neurons each (784-6144($\times$4)-10), FCN2-4 hidden layer network with 600 neurons each (784-600($\times$4)-10). We imported github models \cite{gitxnor, gitbnn} for implementing our experiments. We used the same hyperparameters (such as weight decay value, learning rate etc.) as used in \cite{gitxnor, gitbnn} to train our models. It is worth mentioning that our paper is the only other work besides \cite{galloway2018attacking} demonstrating the effectiveness of discretized/binarized parameter space on adversarial attacks. While \cite{galloway2018attacking} conducted experiments with various forms of attacks (primarily, on MNIST), we restrict ourselves to the WB attack scenario and extrapolate our analysis on larger datasets.

\subsection{Discretization of Input Space}
\setlength{\textfloatsep}{5pt plus 1.0pt minus 4.0pt}
\begin{figure}
\centering
\includegraphics[width=0.85\linewidth]{figures/fig3}
\caption{\textit{Sample images from CIFAR10 dataset for varying levels of input pixel discretization: \textbf{$2-bit, 3-bit, 4-bit, 8-bit$}}}
\end{figure}
With input space discretization, we convert the raw integer pixel intensities ($0 \le I \le 255$) that are typically 8-bit (or $8b$) values to a low precision range of $\delta$ bits ($0 \le I_{\delta} \le 2^\delta$) as:
\begin{equation}
I_\delta = \floor*{\frac{I}{(256/2^\delta)}} (\frac{256}{2^\delta}) + \frac{1}{2} (\frac{256}{2^\delta})
\end{equation}
where $\floor*{}$ denotes integer division. Such quantization reduces the number of data points (given a grayscale input image of size $N\times N$) in the manifold from ${{2^8}^{N^2}}$ to ${{2^\delta}^{N^2}}$. This can be broadly interpreted as reducing the redundancy in the input data. Fig. 3 illustrates sample CIFAR10 images discretized to varying $\delta$ values. The corresponding accuracy (trained on AlexNet for 20 epochs) is shown in Table \ref{table1}. There is a natural tradeoff between input discretization and overall accuracy of a network. Yet, the test accuracy loss from the full precision $8b$ to $3b$ is $\sim2.2\%$. This verifies the presence of unnecessary and redundant input dimensions that do not substantially contribute to the classification task or accuracy. $2b$ discretization decreases the accuracy by a larger margin ($\sim6\%$). Note, this accuracy loss can be minimized by training the $2b$ inputs for more epochs. However, for iso-comparison, we fix the number of epochs across all experiments for a given dataset.
\setlength{\intextsep}{3pt
\setlength{\columnsep}{2pt
\begin{wraptable}{r}{4.5cm}
\setlength{\tabcolsep}{4pt}
\caption{\textbf{CIFAR10 Accuracy}}
\label{table1}
\centering
\begin{tabular}{|c|c|p{\textwidth}}
\hline
Input-bit & Accuracy \\
\hline
2b & 82\\
\hline
3b & 86.64\\
\hline
4b & 87.1\\
\hline
8b & 88.9\\
\hline
\end{tabular}
\end{wraptable}
\setlength{\textfloatsep}{5pt plus 1.0pt minus 4.0pt}
\begin{figure}
\centering
\includegraphics[width=0.7\linewidth]{figures/fig4}
\caption{\textit{Adversarial accuracy on test data for varying perturbation values on (a) CIFAR10 (b) MNIST for different input discretization}}
\end{figure}

A remarkable outcome of this discretization method is the substantial improvement in a model's adversarial accuracy. Fig. 4 illustrates the evolution of adversarial accuracy of the CIFAR10 models (from Table \ref{table1}) with increasing level of perturbation, $\epsilon$. Note, $\epsilon = 0$ corresponds to clean test accuracy. It is clear that clamping the input dimensionality to lower values increases the resistance of the model to larger magnitude of distortion. We can thus deduce that removing the redundant dimensions in the input reduces the overall hyper-volume space thereby leaving 'less' space for shifting or adversarially perturbing a data point (referring to Fig.2 intuition). $8b$ input model shows a decline in accuracy even for a small value of $\epsilon =1/255$. This further confirms that presence of redundant input dimensions allow even small perturbations to shift a data point. In contrast, increasing discretization increases the minimum $\epsilon$ that affects a model's accuracy catastrophically. What is surprising is that for $2b$ input, a model's adversarial accuracy ($\sim79\%$) for large $\epsilon = (8,16)/255$ is almost similar to that of clean accuracy ($\sim82\%$). For larger $\epsilon = 32/255$, the accuracy of all models declines to $<10\%$, except $2b$. This is a very interesting result since we have not employed any adversarial training, and still achieve substantial adversarial resistance for a large $\epsilon$ range

Fig. 4(b) shows the adversarial accuracy results for MNIST (trained on FCN2 for 10 epochs). We observe a similar trend of increasing adversarial resistance with increasing discretization for larger $\epsilon$. Since MNIST is a simple dataset with predominantly black-background, input discretization ($8b,4b,3b$) does not contribute much to adversarial resistance until we go to extremely low $2b$ precision. In fact, $2b$ discretization yields adversarial accuracy similar to the clean test accuracy (for $\epsilon=0.1$) exhibiting the effectiveness of this technique even for simple datasets.

Next, we trained the $2b$ input discretized CIFAR10 and MNIST models with adversarial training to observe the improvement in adversarial accuracy compared to $8b$ input adversarial training (Table \ref{table2}). Comparing to the results in Fig. 4(a, b), adversarial training substantially improves the robustness of a model with full $8b$ input for larger $\epsilon$ values. Input discretization greatly furthers this robustness with $>10\%$ accuracy gain across different perturbation ranges in both MNIST and CIFAR10. It is worth mentioning that the CIFAR10 accuracy (79\%) without adversarial training for $\epsilon =(8,16)/255$ for $2b$ input is as good as the accuracy (83\%) with adversarial training. This shows that input discretization is a good regularization scheme that improves the generalization capability of a network on adversarial data. Note, for $\epsilon =32/255$ in case of CIFAR10, the accuracy is similar for $8b, 2b$ since the adversarial training was conducted with adversaries created using $\epsilon =8/255$. Including larger perturbation adversaries during adversarial training will yield improved accuracy gain.
\begin{table
\caption{\textbf{Accuracy with adversarial training for varying $\epsilon$. Text in \textcolor{red}{red} are the $\epsilon$ values for MNIST and corresponding accuracy.}}
\label{table2}
\centering
\begin{tabular}{|c|c|c|c|c|c|p{\textwidth}}
\hline
Data & Model & Clean & \makecell{$\epsilon$:8/255\\ \textcolor{red}{0.1}} &  \makecell{$\epsilon$:16/255\\ \textcolor{red}{0.2}}&  \makecell{$\epsilon$:32/255\\ \textcolor{red}{0.3}} \\
\hline
\multirow{2}{4em}{CIFAR10}& 2b & 83.1 & 82.7 & 82.7 & 43.9 \\
& 8b & 84.3 & 62.2 & 53.6 & 45.5\\
\hline
\multirow{2}{4em}{MNIST}& 2b & 98.5 & \textcolor{red}{98.5} & \textcolor{red}{84.7} & \textcolor{red}{85.4} \\
& 8b & 98 & \textcolor{red}{84.8} & \textcolor{red}{74.5} & \textcolor{red}{65.9} \\
\hline
\end{tabular}
\end{table}

\subsection{Discretization of Parameter Space}
Since input discretization gave us such promising results, we were naturally inclined toward analyzing a binarized neural network's (BNN) behavior against adversarial attacks. Here, the weights and activations (or parameters) are discretized to extremely low precision values $\{+1, -1\}$ \cite{hubara2016binarized}. The discretization constraints are imposed on a BNN during training, wherein, the parameters are clamped to $\{+1, -1\}$ after every backpropagation step. One can view this discretization as an implicit form of regularization. In fact, it is this extreme form of regularization that makes a BNN difficult to train (clean test accuracy observed with BNNs is, typically, lower than full-precision networks). As suggested in \cite{galloway2018attacking}, the difficulty in training a BNN translates to difficulty in attacking the BNN as well. Referring to the data manifold intuition (Fig. 2), we can deduce that constraining the parameter space during a model's training will introduce discontinuities and non-smoothness in its decision boundary. Since adversaries are created using gradients of a model (that is a property of the model's decision boundary), generating gradients (and hence adversaries) for non-smooth functions will be difficult. This in turn will make a BNN less susceptible to adversaries. Note, the input image to a BNN is full $8b$ precision.
\setlength{\textfloatsep}{3pt plus 1.0pt minus 4.0pt}
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{figures/fig5}
\caption{\textit{Adversarial accuracy on test data for varying perturbation values on (a) CIFAR10 (b) MNIST for binarized and full-precision ($32b$ weights) models}}
\end{figure}

Fig.5(a) compares the adversarial accuracy obtained for varying $\epsilon$ values for CIFAR10 BNN  (AlexNet architecture) against a similar architecture full-precision network (with $32b$ precision for weights and activations). We trained the networks for 40 epochs since BNNs require more training iterations to attain comparable accuracy as that of a full-precision network. Here, we do not incorporate input discretization in our analyses. All networks are fed $8b$ inputs. In Fig. 5(a), for $\epsilon \le 2/255$, BNN shows better adversarial resistance (i.e. adversarial accuracy is closer to clean accuracy). However, the BNN's accuracy declines steeply as we move toward larger perturbation ranges. We note a similar trend for MNIST (trained for 10 epochs on FCN2 architecture), wherein the full-precision network yields improved robustness than the BNN for $\epsilon \ge 0.2$. These results contradict our intuition that increased discretization of BNNs should result in lesser adversarial susceptibility.

To understand this, we calculated the L1 norm of the first hidden layer activation of the FCN2 network in response to clean input images. We found that BNNs generally have a larger variance and range of values than full-precision network. Since BNN uses weight values (+/-1) which are typically of greater magnitude than the small weight values of a full precision network, we observe a larger range in the former case. Interestingly, we find that the L1 norm of the BNN activations (in response to adversarial images perturbed with lower $\epsilon$ values) approximately lie within the same range as that of the clean input case. In contrast, L1 norm for higher $\epsilon$ adversaries have a much higher range. For a full-precision network, the L1 norm range of the different $\epsilon$ adversaries and clean data typically intersect with each other owing to the lower weight values (Fig. 6). We believe that the extreme quantization of weight values in BNNs to higher magnitudes causes adversarial susceptibility for larger range perturbations. While the L1 norm analysis is not very substantial from a mathematical standpoint, it hinted us to increase the capacity (more neurons and weights) of the network. The motif here is that increasing the capacity would increase the overall range of activation values that might incorporate larger range perturbations. Exploding the network capacity for MNIST (FCN1 architecture) yielded a sizable improvement in adversarial resistance with BNN as compared to its corresponding full-precision counterpart (Fig. 5(b)). This is a crucial detail of our analysis that: \textit{while BNNs are intrinsically robust to adversaries (for small $\epsilon$), only models with sufficient capacity can withstand against large $\epsilon$ values}.

\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{figures/fig6}
\caption{\textit{Normalized L1 norm of first hidden layer activations in response to clean ($\epsilon=0$) and adversarial inputs ($\epsilon=0.1, 0.3$) for binarized and full-precision MNIST model of different architecture: FCN1, FCN2}}
\end{figure}

\setlength{\textfloatsep}{3pt plus 0.0pt minus 0.0pt}
\begin{table
\caption{\textbf{Accuracy with adversarial training for varying $\epsilon$. Text in \textcolor{red}{red} are the $\epsilon$ values for MNIST and corresponding accuracy.}}
\label{table3}
\centering
\begin{tabular}{|c|c|c|c|c|c|p{\textwidth}}
\hline
Data & Model & Clean & \makecell{$\epsilon$:8/255\\ \textcolor{red}{0.1}} &  \makecell{$\epsilon$:16/255\\ \textcolor{red}{0.2}}&  \makecell{$\epsilon$:32/255\\ \textcolor{red}{0.3}} \\
\hline
\multirow{2}{4em}{CIFAR10}& BNN & 79.7 & 53.1 & 43.6 & 35.3 \\
& Full & 82.7 & 72.2 & 63.6 & 55.5 \\
\hline
\multirow{2}{4em}{MNIST (FCN1)}& BNN & 96.9 & \textcolor{red}{89.1} & \textcolor{red}{74.5} & \textcolor{red}{65.8} \\
& Full & 98 & \textcolor{red}{84.8} & \textcolor{red}{71} & \textcolor{red}{61.7} \\
\hline
\end{tabular}
\end{table}

Even with adversarial training, we observed the same trend that binarized networks of insufficient capacity do not yield as good adversarial robustness as that of a full-precision network (Table \ref{table3}). For CIFAR10, full-precision network is the clear winner. While for MNIST (with excessive capacity FCN1 architecture), BNN yields improved robustness. A noteworthy observation here is that adversarial training substantially improves the robustness of a full-precision network (see CIFAR10 results in Fig. 5(a) , Table \ref{table3}), while BNNs do not benefit much from them. In fact, we find that BNNs are difficult to train with adversarial training. The learning rate/other hyperparameters need to be tuned carefully to ensure that the BNN model converges to lower error values during adversarial training. \cite{galloway2018attacking} also observed a similar trend and explained that binarized weights are not as \textit{'malleable'} as full-precision weights and hence cannot easily adjust to all possible variations of adversarial data augmented to the training dataset. We think that increasing the capacity of the network compensates for the \textit{`non-malleability'} of the constrained parameters to certain extent. As a result, we see improved accuracy for MNIST in Table \ref{table3} with FCN1 architecture

\subsection{Discretization of Input and Parameter \\Space}
Next, we combined both discretization strategies and analyzed the adversarial robustness of BNNs with varying image-level discretization. We compare the adversarial accuracy of BNNs  to that of a full-precision network for iso-input discretization scenarios, as shown in Table \ref{table4} for CIFAR10 (AlexNet architecture trained for 40 epochs). In Table \ref{table4}, \textit{BNN-2b} (\textit{Full-2b}) refers to a binarized (full-precision) model with $2b$ input precision. Full precision models have $32b$ precision weights and activations. While input discretization for a full-precision network suffers a sizeable accuracy loss, BNN's accuracy fluctuation is marginal with a maximum of $1\%$ change. This is expected since BNNs (owing to +/-1 binarized parameters) do not have as many dimensions (as a full-precision network with $32b$ weights and activations) to fit the extra information in the $8b$ input data. Thus, BNNs fit $2b$, $8b$ data likewise yielding similar generalization error. As opposed to the results seen earlier with $8b$ inputs, BNNs with lower input precision ($2b, 4b$) have significantly higher adversarial resistance than their full-precision counterparts even for large $\epsilon$ values. \textit{Model capacity does not restrict the adversarial resistance in this case. This is an artefact of the two-step quantization that increases the minimum allowable perturbation to shift a data point.} We can also draw an alternate insight from this result: The constrained parameter space of BNNs restricts their overall exploration of the data manifold during training. Referring to Fig. 2 (b), this increases the probability of untrained or arbitrary hyper-volumes (for BNNs) thereby increasing their adversarial susceptibility. Increasing the capacity enables a BNN to explore the manifold better during training. By discretizing the input, we are restricting the overall data manifold space. This allows a model, even, with lower capacity to explore the manifold well thereby decreasing the extent of arbitrary hyper-volumes. Table \ref{table5} illustrates the accuracy results for MNIST (FCN1 architecture trained for 20 epochs).

\begin{table}
\caption{\textbf{Adversarial accuracy with CIFAR10 for varying $\epsilon$ with different combinations of input and parameter discretization.}}
\label{table4}
\centering
\begin{tabular}{|c|c|c|c|c|p{\textwidth}}
\hline
Model & Clean & \makecell{$\epsilon$:8/255} &  \makecell{$\epsilon$:16/255}&  \makecell{$\epsilon$:32/255} \\
\hline
BNN-2b & 81 & 80 & 80 & 36.7 \\
Full-2b & 82 & 79.1 & 79.3 & 39.6 \\
\hline
BNN-4b & 81.9 & 58.3 & 52.3 & 36.8 \\
Full-4b & 81.1 & 53.8 & 45.1 & 37.3 \\
\hline
BNN-8b & 81.5 & 55.1 & 45.9 & 38.6 \\
Full-8b & 86.5 & 61.1 & 55.2 & 47.1 \\
\hline
\end{tabular}
\end{table}

\begin{table}
\caption{\textbf{Adversarial accuracy with MNIST for varying $\epsilon$ with different combinations of input and parameter discretization.}}
\label{table5}
\centering
\begin{tabular}{|c|c|c|c|c|p{\textwidth}}
\hline
Model & Clean & \makecell{$\epsilon$:0.1} &  \makecell{$\epsilon$:0.2}&  \makecell{$\epsilon$:0.3} \\
\hline
BNN-2b & 96.4 & 96.4 & 60.7 & 62.3 \\
Full-2b & 97.8 & 97.4 & 35.4 & 35.3 \\
\hline
BNN-4b & 96.4 & 88.9 & 76.7 & 58.7 \\
Full-4b & 98.1 & 71.1 & 50.9 & 33.7 \\
\hline
BNN-8b & 97.1 & 89.4 & 56.1 & 33.6 \\
Full-8b & 98.2 & 75.9 & 38.5 & 26.4 \\
\hline
\end{tabular}
\end{table}

We conducted adversarial training with $2b$ input discretized BNNs to find out if it helps build adversarial robustness. The results are shown in Table \ref{table6}. Comparing to the $8b$ input BNN adversarial training results in Table \ref{table3}, we observe a substantial gain in adversarial accuracy. However, contrasting the BNN results against Table \ref{table2} ($2b$ input full-precision networks), we observe similar performance gains. In fact, the accuracy gains for $2b$ input CIFAR10 BNN with and without adversarial training (Table\ref{table4}/Table \ref{table6}) are nearly the same. Earlier, we saw that the accuracy (for low $\epsilon$ values) of a full-precision network working on $2b$ input data without adversarial training is similar to that of an adversarially trained network on $8b$ inputs (Table \ref{table2}, Fig. 4). Combining the adversarial training results till now, we can deduce the following: \textit{1) For low input-precision ($2b$) regime, adversarial training does not compound the adversarial resistance of a network (irrespective of binarized or full-precision parameters), for lower $\epsilon$ values.  Adversarial training helps when the input has higher ($8b$) precision. 2) Input discretization, in general, offers very strong adversarial defense  for lower $\epsilon$ values. Discretizing the input as well as the parameter space furthers adversarial robustness. Adversarial training in a discretized input and parameter space does not benefit much and hence can be waived.} However, in case of stronger multi-step attack scenarios and to gain robustness against larger perturbations (such as $\epsilon=32/255, 0.3$ in CIFAR10, MNIST), the network needs to be adversarial trained with corresponding large $\epsilon$ values.
\begin{table}
\caption{\textbf{Accuracy with adversarial training for varying $\epsilon$. Text in \textcolor{red}{red} are the $\epsilon$ values for MNIST and corresponding accuracy.}}
\label{table6}
\centering
\begin{tabular}{|c|c|c|c|c|c|p{\textwidth}}
\hline
Data & Model & Clean & \makecell{$\epsilon$:8/255\\ \textcolor{red}{0.1}} &  \makecell{$\epsilon$:16/255\\ \textcolor{red}{0.2}}&  \makecell{$\epsilon$:32/255\\ \textcolor{red}{0.3}} \\
\hline
CIFAR10& BNN-2b & 78.4 & 78.1 & 78.1 & 30.5 \\
\hline
\makecell{MNIST\\(FCN1)}& BNN-2b & 95.7 & \textcolor{red}{95.7} & \textcolor{red}{89.3} & \textcolor{red}{88.6} \\
\hline
\end{tabular}
\end{table}
\begin{figure}
\centering
\includegraphics[width=0.75\linewidth]{figures/fig7}
\caption{\textit{Adversarial accuracy on test data for increasing $\epsilon$ values on binarized and full-precision ($32b$ weights) models trained on CIFAR100 with different input discretization:\textbf{$8b, 4b, 2b$}}}
\end{figure}
\subsection{Analysis on CIFAR100 and Imagenet}
Scaling up the discretization analysis to larger datasets yielded similar results as observed with CIFAR10, MNIST. Fig. 7 demonstrates the adversarial accuracy evolution for CIFAR100 (trained on ResNet20 architecture for 164 epochs) for binarized XNOR ($1b$ weights and activations) networks and corresponding full-precision ($32b$ weights and activations) models. Note, XNOR networks are similar to BNNs (1-bit weights/activations) with additional scaling factors to achieve higher accuracy on complex datasets. It is evident that input discretization is the most beneficial to obtain adversarial robustness. $2b$ input discretized models in both cases yield adversarial accuracy close to the clean accuracy ($\epsilon =0$) for a large range of perturbations. The accuracy loss between clean and $\epsilon=16/255$ adversary for $2b$-input XNOR ($1.6\%$) is slightly better than the $2b$-input full-precision model ($2.4\%$). This can be attributed to the intrinsic robustness offered by discretizing the parameter space of XNOR networks. Furthermore, the fact that $8b$-input XNOR yields higher adversarial accuracy for iso-perturbation values than $8b$-input full-precision model further demonstrates the ability of binarized networks to counter adversarial attacks. A noteworthy observation here is that the loss in clean accuracy between $2b$ and $8b$ input discretized full-precision network is small ($\sim2\%$) as compared to the large $6\%$ loss observed earlier with CIFAR10 (Table \ref{table1}). As we scale up the complexity of the dataset, the redundancies in the input dimensions increase. Discretizing the input for a complex dataset thus eliminates more redundant dimensions that do not contribute to the accuracy. In contrast, smaller datasets have lesser redundant dimensions and are thus at a risk of suffering a large accuracy drop with input discretization.
\setlength{\textfloatsep}{5pt plus 1.0pt minus 4.0pt}
\begin{figure}
\centering
\includegraphics[width=0.75\linewidth]{figures/fig8}
\caption{\textit{Top-5 Adversarial accuracy on test data for increasing $\epsilon$ values on binarized and full-precision ($32b$ weights) models trained on Imagenet with different input discretization:\textbf{$8b, 2b$}}}
\end{figure}

Fig. 8 shows the accuracy results for Imagenet (trained on AlexNet). We only show the top-5 adversarial accuracy. We see similar trends as CIFAR100. Note, the XNOR models are trained for 50 epochs, while full-precision models are trained for 90 epochs. As a result, we see lower baseline accuracy ($\epsilon=0$) in the former case. Like CIFAR100, the loss in clean test accuracy between $2b$ and $8b$ input discretization is minimal for each model. Also, the accuracy difference between clean and $\epsilon=16/255$ adversarial data for $2b-XNOR$ ($0.9\%$) is much lower than $2b-full$ precision models ($2.8\%$). This highlights the intrinsic robustness capability of binarized networks even for large-scale datasets.

\section{Conclusion}

Low-precision models or quantization techniques, so far, have been explored to reduce the resource utilization of DLNs for energy-efficient deployment on edge devices. We have demonstrated that \textit{discretization} also warrants \textit{security} against adversarial attacks, thereby, offering a key benefit of \textit{robustness} in hardware implementation. In summary, the main findings/recommendations from this work are:
\begin{itemize}
\item{Input discretization is major benefactor for adversarial robustness (with both binarized and full-precision) models. $2b$ input discretized models (without adversarial training) yield similar adversarial accuracy as adversarially trained $8b$ input models for lower $\epsilon$ values. Robustness against higher $\epsilon$ and multi-step attack requires adversarial training.}
\item{Binarized (low-precision weights/activations) models are intrinsically more (although, slight) robust than full-precision ($32b$ weights/activations) models. Adversarial training needs to be carefully done on sufficient capacity binarized networks to attain similar adversarial robustness as the full-precision models.}
\item{Combining input and parameter discretization is an efficient way of obtaining adversarial robustness to a moderate range of perturbation values without conducting the iterative adversarial training.}
\end{itemize}
Our work unravels a simple idea that: \textit{hardware optimization related techniques can potentially resolve or resist software vulnerabilities (specifically, adversarial attacks)}. While we focus on discretization, there is a lot of future scope to explore other efficiency-driven techniques (such as, stochasticity, model pruning etc.) to gauge their implication on adversarial robustness.

\section*{Acknowledgment}
This work was supported in part by C-BRIC, Center for Brain-inspired Computing, a JUMP center sponsored by DARPA and SRC, by the Semiconductor Research Corporation, the National Science Foundation, Intel Corporation and by the Vannevar Bush Faculty Fellowship.

redpajama_set_name: RedPajamaArXiv


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Windows XP was an Operating System for the PC by Microsoft. It was published in 2002 and is now obsolete, since it is no longer supported with updates.
Building on the success of Windows 2000, XP offered many improvements, especially for gamers. Compatibility for older games was dramatically improved by use of the new Compatibility Mode. Still, since XP (as well as the older Windows 2000) was released when the Classic-Tomb Raider Games were already on the market, in some cases they are not compatible with it, meaning that the game might crash after being started or not start at all. There are, however, many third party Patches addressing these problems. Many newer games that were released after Microsoft stopped supporting XP, still run on XP, though it is not recommended.
Remark: This article describes Compatibility of the original game discs with Windows. For information on newer versions (like downloadable Steam games), see Steam.
Windows NT and newer systems based on it do not support real mode DOS.
On 32-bit machines Tomb Raider can by played in the command shell, although many limitations do apply, like problems with sound and only low resolution graphics without hardware acceleration.
On 64-bit versions of Windows it is technically impossible to run 16-bit applications like the original Tomb Raider in the command shell. You will have to use Glidos or DosBox.
Tomb Raider II through Chronicles will for the most part run at once or with the "Windows 98/ME" compatibility mode enabled that has been available since Windows XP.
From Vista on Microsoft has removed the Direct3D interface used by EAX. Owners of a Sound Blaster X-Fi sound card manufactured by Creative can avoid this problem by using the program ALchemy. You might also try the Multi Patch listed at tombraiderchronicles.com.
Officially supported only on Windows 7 and newer. Needs at least DirectX 11.
This page was last modified on 8 November 2017, at 17:42.


text:
Topic Cloud computing
CIO Decisions
Taking control with cloud inventory management
NewsMidmarket CIOs: A new IT career path to wealth?
FeatureCloud inventory management for CIOs: Build guardrails, not roadblocks
FeatureProfiles in digital leadership at Forbes, CBS, US Bank
FeatureNYT CIO uses niche apps to crack the digital media code
ColumnCloud inventory management: A 21st-century CIO's responsibility
everythingpossible - Fotolia
Cloud inventory management for CIOs: Build guardrails, not roadblocks
Cloud inventory management saves money, reduces security risk and helps the enterprise compete. But first, CIOs must build relationships with business decision makers.
Mary K. Pratt
Like most IT leaders, CIO Bryce Austin has contended with shadow IT for years.
But today, instead of tracking down hidden servers to find rogue technology, Austin focuses mainly on cloud inventory management.
"Cloud makes it easier to bypass IT for larger functions," said Austin, CIO at consulting firm Digineer.
Individual workers and business groups alike can power up a new application easily and quickly without IT ever knowing, he said. Tech vendors increasingly go straight to business unit leaders with their sales pitches, raising the likelihood of someone deploying cloud apps without IT's blessing.
The business' easy access to cloud apps and vendors might seem to make the IT function obsolete, but Austin and others said the lack of cloud inventory management makes the CIO role more critical in many ways, because CIOs and their staff must now find and inventory such cloud apps to determine whether and when they make sense for the organization. Indeed, the situation actually further shifts how CIOs do their job and even how they should think about rogue IT.
Chris Curran
"Shadow IT never went away and it's not a negative. It just is. It's the way we do and will work," said Chris Curran, a PwC principal and chief technologist for the U.S. firm's advisory practice. "People who need information and tools should be able to get them whenever they need them. But the issue is doing it in the context of the information, security, privacy and integration protection that's needed."
Data security, pricing jeopardized by rogue cloud apps
Organizations of all kinds are encountering workers using cloud apps without IT's knowledge. The usage stems from both individual workers seeking out cloud apps to help them perform a particular task, as well as entire departments lighting up enterprise apps in the cloud, said Forrester Research analyst Lauren Nelson. It's easy to do in both cases and often creates efficiencies in business processes for the workers and departments involved.
Unfortunately, many CIOs are left out of the loop, and as a result, they quickly lose track of what apps are performing which functions, Nelson said. "You think you've identified what's being used, but then you find there are people using apps that didn't go through your process."
Nelson said she worked with one company that, after finding 600 user accounts for cloud apps, went into firefighting mode, eliminating high-risk items where there were security concerns only to find at the end of that process the updated cloud inventory was out of date again.
The role of the CIO has changed. People used to come to you and you were the king of the hill. ... Now it's very much the other way around. Now you're the sales guy, the marketing guy.
Scott YoungsCIO, Key Information Systems
So, why even attempt cloud inventory management?
Data flowing through rogue cloud apps -- or actually any unsanctioned app, cloud or on premises -- increases risks around security and regulatory requirements, Nelson and others said. In fact, a 2014 study from cloud security company Netskope estimated that the use of cloud services by the business increases the likelihood of a data breach threefold.
Additionally, with data flowing through rogue cloud apps, experts said a company could lose the so-called single version of truth for its data, a key component of corporate analytics programs and decision making.
Then there are cost factors. An organization with multiple independent accounts for the same cloud service or cloud app is hardly going to get the best rate. It may also be paying for duplicate and redundant services.
CIOs recognize these potential problems, but Nelson said most still struggle to address the situation.
Emerging best practices for cloud inventory management
Nelson, Curran and others said there's no one tool -- no silver bullet -- that a CIO can deploy to track and inventory all the cloud applications that are being engaged by employees and departments.
"What's more typical, you see people collecting and building spreadsheets and doing analysis on their own," Curran said.
There are, however, emerging best practices for how to track and inventory what's happening in the cloud.
Nelson said implementing a cloud inventory management strategy should start with the CIO and IT staff recognizing that what's happening isn't necessarily a bad thing. "These are innovative employees who are trying to find tools that bring value and differentiation," she said. "So, [CIOs] want to provide guardrails without stomping it out."
Armed with that mentality, CIOs should implement or, if already in place, better employ tools (i.e., network traffic analyzers and cloud management tools) to scan traffic to see where data is going and whether it's going to cloud services, Nelson said.
Then CIOs should ask employees to let them know if they're using cloud apps, she and others said. After that, CIOs should get in front of the situation, so workers seeking apps can approach IT for advice or, perhaps, select from a list of vendors and products already approved by IT.
Scott Youngs
Scott Youngs, the CIO at Key Information Systems, an IT services provider, acknowledged the challenge in tracking and keeping an up-to-date inventory.
"If they're going outside, there's no way for me to know that they've done that unless I go ask them," he said, adding that even monitoring tools won't catch every cloud app "unless I know every cloud IP address."
He added: "You can't stop it. The best thing you can do is have relationships with the lines of business."
Addressing rogue cloud apps at line-manager level
But Youngs said he and other CIOs can't just build relationships with other executives if they want to track what's going on.
"I have to go further down than I used to; it's first-line managers, too. You create those relationships so people will come to you," he said. "The role of the CIO has changed. People used to come to you and you were the king of the hill. You'd dole out whatever resources you felt were necessary and it was very much the lines of business come to the mountain. Now it's very much the other way around. Now you're the sales guy, the marketing guy. I have to go out to every department and ask, 'Do you have what you need?' because people want to move as fast as they can, and you cannot be the bottleneck."
Curran agreed: "It's about advising and guiding and integrating and less about controlling. The role changes from control to integration and visibility."
In fact, Curran recommended abandoning the idea of getting and working from a full written cloud inventory anyway; he said trying to collect this with the idea of finding someone skirting the system "creates a contentious environment."
Instead, he said the best approach is to build relationships, "to communicate what you're trying to do, why you're doing it and what's the benefit for the organization."
Austin, the Digineer CIO, said he learned that lesson.
He said he learned that his sales department had gotten a cloud-based analytics app when it sought his help integrating the app with back-end systems. Sure, the department had good reason for getting the app, he said, but noted that IT could have helped with the procurement process to ensure the selected application was the best value for the company in all regards.
But Austin takes the blame for sales going around IT. "I hadn't developed a strong enough relationship with the head of sales that he thought to come to IT for the best decision," he said.
Stemming the need for shadow cloud
Now, Austin said he tries to track rogue cloud apps in several ways. He said he works with the accounts payable people to know what tech vendors are getting corporate checks, and he works with his IT staff to monitor traffic moving between corporate computers and websites via the internal network. He said he also gets suspicious if his IT team sees a significant drop-off in activity on longstanding corporate applications.
More importantly, Austin said he's working to stem the need for rogue cloud apps. He's giving workers and department leaders more options so they know they can come to his IT department first for what they need.
"The best way to make sure it doesn't happen is to offer better products and services from my own organization than the things that the people in my company are seeking out in the cloud," he said.
CIOs will see a real ROI on taking these approaches. Beyond being able to track and get that sought-after cloud inventory, CIOs will be able to better ensure data security and better control costs.
Moreover, CIOs will be able to keep themselves relevant and critical for corporate success.
"Right now, there is more and more pressure to differentiate your customer service, and if central IT can't give [employees] the tools they need at the time they need them, the business will get it themselves. So, IT has to learn to do things differently or they'll be in charge of only back-end systems and not the innovative systems," Nelson said.
Article 2 of 5Next Article
Watch this video presentation from Derek Lonsdale, IT transformation leader and Lean expert at PA Consulting, on how understanding business processes plays into cloud inventory management. Here, CIOs explain why cloud economics will differ for each business.
This was last published in July 2015
CW+: Study: The economic benefits of cloud to business and the economy –ComputerWeekly.com
E-Guide: How to Identify Which Applications are Right for Public Cloud Computing –SearchDataCenter.com
Cloud Computing for Business –ComputerWeekly.com
Private Cloud Computing E-Zine Volume 1: Private cloud computing as a business ... –SearchDataCenter.com
Mary K. Pratt asks:
What's the biggest mistake CIOs and IT departments make with cloud inventory management?
Mary Pratt - 17 Jul 2015 7:03 AM
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redpajama_set_name: RedPajamaCommonCrawl


A sports mind with a very inqisitive nature. For example, why am I writing this bio? That was clearly a joke. Oh, I may be funny as well. The jury is still out on that one. Sports are always first on my mind and if you need a chuckle well, you just may get one here.


text:
Roger Keating
Senior Vice President, Digital Media, Hearst Television
Roger Keating runs the digital media effort for Hearst Television. His scope includes Hearst Television's station sites, mobile apps, social expressions, OTT and advanced advertising. He was a co-founder and board member of Pearl, a venture formed by 9 of the largest local TV groups in the country to pursue opportunities requiring scale and ubiquity; of Mobile Content Ventures, a partnership between Pearl, NBCU, Fox and Ion to deliver the "Dyle" mobile television service. And of NewsON, a venture committed to aggregating the video newscast content from stations around the country into a single set of mobile and connected TV apps.
Earlier in his career, Roger ran large cable TV operations, first at Comcast and then at Time Warner where he presided over the integration of the Adelphia acquisition in the Los Angeles Region. He previously ran the voice services business at AOL, as well as the AOL Health and AOL Travel verticals. He led the formation of Comcast's Internet division in the mid 90's and oversaw the development of their high speed data business. And he was founder/CEO of a dot com in 1998 named "Zatso" that worked with local TV broadcasters to offer streaming video-based "personalized newscasts."
All session by Roger Keating
THE FUTURE OF OTT: CONSUMER ADOPTION AND EXPERIENCES


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This last Thursday, March 3rd, we delivered 29 bicycles to Tollgate Elementary to help them start a bike club for 3rd graders. Getting bikes into the hands of these children is one of the most rewarding aspects of Bikes Together. The bicycle can open up an entire new world for kids that can't be replicated elsewhere. Getting someone to be a dedicated cyclist as an adult can start with planting that seed as a child because the bicycle can be an important tool in development.
Strength and Coordination – Riding a bike helps kids develop a sense of balance, working the legs and the core, making strong muscles for future growth.
Heart and Lungs – Kids who exercise regularly stay healthy, avoid childhood obesity and set themselves up for health success as an adult.
Freedom and Independence – Without a doubt, magic happens when you push your pedals and go forward on a bicycle. Children can develop a strong sense of independence – a hugely important thing to learn – on a bicycle. For the first time they can power themselves to where they want to go. They can make plans to meet friends, bike to parks, explore their neighborhood and gain confidence to face the world.
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text:
Q: How do I put two label together? I am doing a UI and using tkinter to establish it. As seen in the picture below I want the two labels above equally.

This is the code that I have written for creating this.
Player1 = "Terry"
Player2 = "David"
player1 = tk.Label(top,
                   text=Player1,
                   font="Times 45",
                   fg="white",
                   bg="#0000FF")
player1.pack(side=TOP, padx=10,pady=10,  anchor=NW)
player2 = tk.Label(top,
                   text=Player2,
                   font="Times 45",
                   fg="white",
                   bg="#FF0000")
player2.pack(side=TOP, padx=10,pady=10,anchor=N)

I tried to adjust it using anchor and side but form reason I cant make the other label go up.

A: grid is an alternative to pack that might be easier in this case. Essentially, your interface is divided into a grid of rows and columns, and locations of objects are specified by their row and column location. Your code might look like this:
player1.grid(row=0, column=0)
player2.grid(row=0, column=1)

If you'd rather stick with pack, you just need to specify side as 'left'/'right':
player1.pack(side='left')
player2.pack(side='left`)

A: If you are expecting the output as below, then here is your code.

Player1 = "Terrry"
Player2 = "David"
player1 = tk.Label(top,
                   text=Player1,
                   font="Times 45",
                   fg="white",
                   bg="#0000FF")
player1.pack(side=tk.LEFT, padx=10,pady=10, anchor=tk.NW)
player2 = tk.Label(top,
                   text=Player2,
                   font="Times 45",
                   fg="white",
                   bg="#FF0000")
player2.pack(side=tk.LEFT, padx=10,pady=10, anchor=tk.N)

Instead of passing 'TOP' to the 'side' attribute you could pass 'LEFT' to achieve this. Also, you could either use 'NW' or 'N' to the 'anchor' attribute.

redpajama_set_name: RedPajamaStackExchange


text:
Volumes > Volume 13 > No. 3
Effect of Inertia on Viscous Fluid in a wall-driven Corner Flow with Leakage, Volume 13, No. 3 (May 2019), PP:353-360
Asif Mahmood - Department of Mathematics, Pennsylvania State University, York Campus, 1031 Edgecomb Avenue, York, PA 17403, USA
Predator-Two Preys Population Dynamics Model with Adaptive Management, Volume 13, No. 3 (May 2019), PP:361-367
J. J. Yermekbayeva - Department of Systems Analysis and Control, L.N.Gumilyov Eurasian National University, Astana, Kazakhstan
A. N.Omarov - Department of Systems Analysis and Control, L.N.Gumilyov Eurasian National University, Astana, Kazakhstan
L. V. Nedorezov - Research Center for Interdisciplinary Environmental Cooperation RAS, Saint-Petersburg, Russia
R. E.Shakirova - Department of Systems Analysis and Control, L.N.Gumilyov Eurasian National University, Astana, Kazakhstan
Ostrowski-Type Inequalities for Functions Whose Derivative Modulus is Relatively (m, h1, h2)−Convex., Volume 13, No. 3 (May 2019), PP:369-378
Miguel Vivas-Cortez - Pontificia Universidad Cat´olica del Ecuador , Facultad de Ciencias Exactas y Naturales, Escuela de Ciencias F´ısicas y Matem´atica, Av. 12 de octubre 1076, Apartado:17-01-2184,Telf. 2991700 ext. 2036, Quito, Ecuador.
Carlos Garc´ıa - Universidad Centroccidental Lisandro Alvarado, Decanato de Ciencias y Tecnolog´ıa, Departamento de Matem´aticas, Barquisimeto, Venezuela.
Jorge Eliecer Hernandez H. - Universidad Centroccidental Lisandro Alvarado, Decanato de Ciencias Econ´omicas y Empresariales, Departamento de T´ecnicas Cuantitativas, Barquisimeto, Venezuela
A Model-Driven Engineering Approach for Automating the Portability of User Interfaces in Native Mobile Applications, Volume 13, No. 3 (May 2019), PP:379-391
Riham Abdel Kader - Department of Mathematics and Computer Science, Beirut Arab University, Beirut, Lebanon
Wassim El Hajj Chehade - Department of Mathematics and Computer Science, Beirut Arab University, Beirut, Lebanon
Prediction of Spread Shear Strength of Rock with Ordinary Point Kriging Method using GStat-R, Volume 13, No. 3 (May 2019), PP:393-399
Atje Setiawan Abdullah - Department of Computer Science, Universitas Padjadjaran, Indonesia
Nur Hamid - Department of Geology, Sekolah Tinggi Teknologi Mineral, Indonesia
Annisa Nur Falah - Department of Mathematics, Universitas Padjadjaran, Indonesia
Budi Nurani Ruchjana - Department of Mathematics and Indonesian Ethnoscience Studies Center, Universitas Padjadjaran, Indonesia
The Study of Asymptotic Behavior of Positive Solutions and its Stability for a New Class of Hyperbolic Differential System, Volume 13, No. 3 (May 2019), PP:341-349
Mohammed Said Touati Brahim - Department of Mathematics and Computer Science, Larbi Tebessi University, 12002, Tebessa, Algeria
Tarek Abdulkafi Alloush - Department of Mathematics, College of Sciences and Arts, Al-Rass, Qassim University, Kingdom of Saudi Arabia.
Bahri Belgacem Cherif - Institut preparatoire aux etudes d'ingenieur a Sfax (I.P.E.I.S.Sfax), BP 1172, Sfax, 3018, Tunisie
Ahmed Himadan Ahmed - Department of Mathematics, College of Sciences and Arts, Al-Rass, Qassim University, Kingdom of Saudi Arabia
A Single-Product Stochastic Inventory System Modulated by A Queueing System and Subject to A Compulsory Waiting Period for Re-Ordering, Volume 13, No. 3 (May 2019), PP:351-359
Venkata S. S. Yadavalli - Department of Industrial & Systems Engineering, University of Pretoria, South Africa
Some WSN Characteristics Effects on the Performance of LEACH and MODLEACH Protocols, Volume 13, No. 3 (May 2019), PP:361-368
Hamdy H. El-Sayed - Mathematics Department, Faculty of Science, Sohag University, Egypt
Production, Capacity and Workforce Planning: A Mathematical Model Approach, Volume 13, No. 3 (May 2019), PP:369-382
M. Sivasundari - Department of Industrial Engineering, Anna University, Guindy 600025, Chennai, India
K. Suryaprakasa Rao - Department of Industrial Engineering, Anna University, Guindy 600025, Chennai, India
R. Raju - Department of Industrial Engineering, Anna University, Guindy 600025, Chennai, India
Mathematical Modeling and Quantitative Feedback design for Industrial Nonlinear Storage Tank Level Control (NSTLC) Process, Volume 13, No. 3 (May 2019), PP:383-389
T. Rajesh - Department of Electronics and Instrumentation Engineering, Bannari Amman Institute of Technology Sathyamangalam, Erode, Tamil Nadu, India, 638401 Anna University, Tamilnadu, India
M. Tamilselvan - Department of Electronics and Instrumentation Engineering, Bannari Amman Institute of Technology Sathyamangalam, Erode, Tamil Nadu, India, 638401 Anna University, Tamilnadu, India
An Efficient Packet Loss Resilience Method Using Closed-Form Solution for Unequal FEC Assignment, Volume 13, No. 3 (May 2019), PP:391-400
K. Sivakumar - Department of Information and Communication, Anna University, Chennai, Tamil Nadu 600025, India
S. Sasikumar - Department of Electronics and Communication Engineering, Hindustan Institute of Technology and Science, Chennai, Tamil Nadu 603103, India
Secure and Efficient Deduplication Scheme based on Ownership Challenge for Mobile Cloud Environment, Volume 13, No. 3 (May 2019), PP:401-410
Sebastian Annie Joice - Department of Computer Science and Engineering, Government College of Engineering,Srirangam,Tiruchirappalli, India
M. A. Maluk Mohamed - Department of Computer Science and Engineering, M.A.M College of Engineering,Tiruchirappalli, India
On Generalized Strongly Convex Functions Involving Bifunction, Volume 13, No. 3 (May 2019), PP:411-416
Muhammad Aslam Noor - Mathematics Department, COMSATS University Islamabad, Park Road, Islamabad, Pakistan
Khalida Inayat Noor - Mathematics Department, COMSATS University Islamabad, Park Road, Islamabad, Pakistan
Optimal Oscillation Conditions for a Delay Differential Equation, Volume 13, No. 3 (May 2019), PP:417-425
I. P. Stavroulakis - Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece
Zh. Kh. Zhunussova - Faculty of Mechanics and Mathematics, Al-Farabi Kazakh National University, Almaty, 050040, Kazakhstan
S. Sh. Ixanov - Faculty of Information Technology, Al-Farabi Kazakh National University, Almaty, 050040, Kazakhstan
Belal S. H. Rababh - Department of Business Administration, College of Administrative Sciences, Applied Science University, Kingdom of Bahrain
Numerical Modeling of Heat and Mass Transfer with a Single-Phase Flow in a Porous Cavity, Volume 13, No. 3 (May 2019), PP:427-435
Mohamed F. El-Amin - College of Engineering, Effat University, Jeddah 21478, KSA
Shereen Abdel-Naeem - Mathematics Department, Faculty of Science, Aswan University, Aswan 81528, Egypt
Nehma A. Ebrahiem - Mathematics Department, Faculty of Science, Aswan University, Aswan 81528, Egypt
Euclidean Jordan Algebras, Strongly Regular Graphs and Cauchy Schwarz Inequalities, Volume 13, No. 3 (May 2019), PP:437-444
Lu´ıs Antonio de Almeida Vieira - Section of Mathematics, Department of C´ıvil Engineering, University of Porto, Street Dr Roberto Frias, 0351-4200-465 Porto, Portugal
Scheduling and Load Balancing using NAERR in Cloud Computing Environment, Volume 13, No. 3 (May 2019), PP:445-451
J. Arul Sindiya - Department of Computer Science and Engineering, CARE Group of Institutions, Trichy,Tamil nadu India.
R. Pushpalakshmi - Department of Information Technology, PSNA College of Engineering and Technology, Dindigul,Tamil nadu India
An Extended Model of Penalized Spline with The Addition of Kernel Functions in Nonparametric Regression Model, Volume 13, No. 3 (May 2019), PP:453-460
Rahmat Hidayat - Department of Mathematics, Faculty of Science, Cokroaminoto Palopo University, South Sulawesi/ 91913, Indonesia
I. Nyoman Budiantara - Department of Statistics, Faculty of Mathematics, Computation and Data Science, Institut Teknologi Sepuluh Nopember, Surabaya/ 60111, Indonesia
Bambang W. Otok - Department of Statistics, Faculty of Mathematics, Computation and Data Science, Institut Teknologi Sepuluh Nopember, Surabaya/ 60111, Indonesia
Vita Ratnasari - Department of Statistics, Faculty of Mathematics, Computation and Data Science, Institut Teknologi Sepuluh Nopember, Surabaya/ 60111, Indonesia
Investigation of Accelerated Aging Effects in Phenolic Ablative Composites, Volume 13, No. 3 (May 2019), PP:461-469
D. S. Manoj Abraham - Anna University, Chennai - 600025, Tamilnadu, India
H. Kanagasabapathy - Dept of Mechanical Engg, National Engineering College, Kovilpatti-628501, India
R. Joselin - Dept of Mechanical Engg, C.S.I.Institute of Technology, Thovalai-629302, India
The Narrative of Renewable Sources of Energy in Science Fiction Films, Volume 13, No. 3 (May 2019), PP:471-480
Islam Abohela - Civil and Architectural Engineering Department, College of Engineering, Applied Science University, Bahrain
Autocorrelation Analysis of Gamma Ray Log of Ras Fanar Oil Field in Egypt, Volume 13, No. 3 (May 2019), PP:481-487
A. Al-Sayed - Physics Department, Faculty of Science, Zagazig University, 44519 Sharqia, Egypt
Atef El-Taher - Physics Department, Faculty of Science, Al-Azhar University, Assuit Branch, 71542 Assuit, Egypt
K. S. Al-Mugren - Physics Department, Faculty of Science, Princess Nourah Bint AbdulRahman University, Riyadh, KSA
An Effective Self-Adaptive Policy for Optimal Video Quality over Heterogeneous Mobile Devices and Network Discovery Services, Volume 13, No. 3 (May 2019), PP:489-505
Saleh Ali Alomari - Faculty of Science and Information Technology, Jadara University, 21110 Irbid, Jordan
Mowafaq Salem Alzboon - Faculty of Science and Information Technology, Jadara University, 21110 Irbid, Jordan
Belal Zaqaibeh - Faculty of Science and Information Technology, Jadara University, 21110 Irbid, Jordan
Mohammad Subhi Al-Batah - Faculty of Science and Information Technology, Jadara University, 21110 Irbid, Jordan
Quintic Non-polynomial Spline Method for Solving the Time Fractional Biharmonic Equation, Volume 13, No. 3 (May 2019), PP:507-513
Adel R. Hadhoud - Department of Mathematics and Computer Science, Faculty of Science, Menoufia University, Egypt


Dau Nguon Resort feels miles away despite being just under three kilometres from the centre.
Follow Tran Nhat Duat Street northwest past the Buon Ako Dhong tourist village (you'll see the traditional long houses) and eventually you will end up at a small pond with brick bungalows on the hillside. When we visited the property was very quiet and after some poking around, we did eventually find someone to show us the rooms. The bungalows are surprisingly cute and cheerful inside. The exposed brick walls have been painted white making everything bright and the big windows also let in a lot of light. A double bed is pushed into one corner and on top of clean tile floors, there's a desk and minifridge. Bungalows come with fan, air-con, kettle, WiFi and an attached wet-room style bathroom with hot water shower. The only disappointing thing is that there's no attached outdoor terrace to enjoy the quiet surrounds. One option is to pop down to one of the many huts surrounding the pond, which we suspect attract small gatherings on the weekends. The place also has one large-ish restaurant dining terrace overlooking the water.
This is definitely a place to chill out, do nothing and pretty much be left alone, not to be social. No English spoken. If you don't have your own motorbike, a one-way taxi ride will cost 40,000-50,000 dong.
Check rates at Dau Nguon Resort on Booking.com.


text:
package de.johmat.myjojox.utils.stream.object;

import java.io.EOFException;
import java.io.IOException;
import java.io.InputStream;
import java.lang.instrument.IllegalClassFormatException;
import java.util.HashMap;
import java.util.Map;

import com.google.gson.JsonSyntaxException;

import de.johmat.myjojox.utils.data.DataConversionUtils;
import de.johmat.myjojox.utils.data.json.JSONUtils;
import de.johmat.myjojox.utils.data.json.ObjectInvalidException;
import de.johmat.myjojox.utils.list.Queue;
import de.johmat.myjojox.utils.log.Logger;
import de.johmat.myjojox.utils.reflect.InheritanceUtils;
import de.johmat.myjojox.utils.stream.impl.StreamValues;
import de.johmat.myjojox.utils.stream.object.data.InvalidObjectContainer;
import de.johmat.myjojox.utils.stream.object.exceptions.DeserializationFailed;
import de.johmat.myjojox.utils.stream.object.exceptions.WrongMagicNumberException;
import de.johmat.myjojox.utils.stream.object.exceptions.WrongStreamVersion;

/**
 *
 * @author MyJojoX
 *
 * @version 1.0
 */
public class ObjectInputStream {

	private InputStream inNative;
	private Queue<Object> queue;
	private boolean closed = false;
	private Map<Integer, String> shortened;
	private boolean block = false;
	private Object blockSync = new Object();

	public ObjectInputStream(InputStream inputStream) throws IOException {
		this.inNative = inputStream;

		byte m1 = (byte) inNative.read();
		byte m2 = (byte) inNative.read();

		if (m1 != ObjectStreamMagicPacket.MAGIC_1 ||
				m2 != ObjectStreamMagicPacket.MAGIC_2)
			throw new WrongMagicNumberException(m1, m2);

		queue = new Queue<>();
		shortened = new HashMap<>();

		fetch();

		if (queue.isEmpty())
			throw new IllegalStateException("did not received magic packet");

		ObjectStreamMagicPacket magic = (ObjectStreamMagicPacket) queue.get();

		if (!magic.isStreamValid()) {
			throw new WrongStreamVersion();
		}
	}

	public void close() throws IOException {
		closed = true;
		inNative.close();
	}

	public void closeSafe() {
		try {
			close();
		} catch (IOException e) {
			Logger.getInstance().error(e);
		}
	}

	@SuppressWarnings("unchecked")
	public <T> T readObject(Class<T> clzz, boolean autofetch) throws IOException {
		Object obj = readObject(autofetch);
		if (obj != null && InheritanceUtils.distance(obj.getClass(), clzz) > -1) {
			return (T) obj;
		} else {
			throw new IllegalStateException(new IllegalClassFormatException("read "+(obj!=null?obj.getClass():"null")+" !instanceof "+clzz));
		}
	}

	public <T> T readObject(Class<T> clzz) throws IOException {
		return readObject(clzz, true);
	}

	public Object readObject() throws IOException {
		return readObject(true);
	}

	synchronized public Object readObject(boolean autofetch) throws IOException {
		if (autofetch && queue.isEmpty()) {
			while (fetch())
				Thread.yield();
		}
		return queue.get();
	}

	public boolean fetch() throws IOException {
		synchronized (blockSync) {
			while (block) {
				if (closed)
					new IllegalStateException("stream was closed");
				try {
					Thread.sleep(10);
				} catch (InterruptedException e) {}
			}

			if (closed)
				new IllegalStateException("stream was closed");

			byte[] intBuffer = new byte[Integer.BYTES];

			inNative.read(intBuffer);
			if (DataConversionUtils.bytesToInt(intBuffer) > StreamValues.MAX_STREAM_PACKET_SIZE)
				throw new OutOfMemoryError("packet size exceeded max packet size");
			byte[] classNameBuffer = new byte[DataConversionUtils.bytesToInt(intBuffer)];
			inNative.read(classNameBuffer);
			String className = getClassName(DataConversionUtils.bytesToStr(classNameBuffer));

			if (classNameBuffer.length == 0)
				throw new EOFException();

			inNative.read(intBuffer);
			if (DataConversionUtils.bytesToInt(intBuffer) > StreamValues.MAX_STREAM_PACKET_SIZE)
				throw new OutOfMemoryError("packet size exceeded max packet size");
			byte[] contentBuffer = new byte[DataConversionUtils.bytesToInt(intBuffer)];
			inNative.read(contentBuffer);
			String content = DataConversionUtils.bytesToStr(contentBuffer);

			try {
				Class<?> clzz = Class.forName(className);
				if (Object.class.isAssignableFrom(clzz)) {
					try {
						Object obj = null;
						boolean invalid = false;

						try {
							obj = JSONUtils.fromJSON(content, clzz);
						} catch (ObjectInvalidException e) {
							if (e.getFailedObject() == null) {
								throw new DeserializationFailed("failed to deserialize "+className);
							} else {
								obj = e.getFailedObject();
								invalid = true;
							}
//							TODO WARN
						}

						if (obj instanceof ClassNameShortenerPacket) {

							ClassNameShortenerPacket pack = (ClassNameShortenerPacket) obj;
							shortened.put(pack.getNum(), pack.getClassName());

							return true;
						} else {
							if (!invalid) {
								queue.put(obj);
							} else {
								queue.put(new InvalidObjectContainer(obj));
							}
						}

					} catch (JsonSyntaxException e) {
						throw new DeserializationFailed(e, className+"$json: "+content);
					}
				} else {
					throw new DeserializationFailed(""+className+" not instanceof JSONSerializable");
				}
			} catch (ClassNotFoundException e) {
				throw new DeserializationFailed(e, className+" not found");
			}
			return false;
		}
	}

	private String getClassName(String name) {
		if (name.startsWith("%"))
			return shortened.get(Integer.valueOf(name.substring(1)));
		else
			return name;
	}

	public InputStream getNativeIn() {
		return inNative;
	}

	public void setNativeIn(InputStream inNative) {
		if (!block)
			throw new IllegalStateException("stream not blocked");
		this.inNative = inNative;
	}

	public void block() {
		if (block)
			return;
		synchronized (blockSync) {
			block = true;
		}
	}

	public void unblock() {
		block = false;
	}

}

redpajama_set_name: RedPajamaGithub


text:
Prevalence of Bertolotti's Syndrome in Lumbosacral Surgery Procedures Research Article
The London Spine Unit : most advanced treatment clinic in London
Introduction Bertolotti's syndrome (BS) describes the relationship between low back pain (LBP) and lumbosacral transitional vertebra (LSTV). It is a factor that is sometimes overlooked when it comes to evaluating and treating LBP.Because of the different diagnostic modalities and criteria used in the research, the LSTV incidence in the general population varies greatly, and hence the link between LSTV and LBP remains contentious.^( )Some researchers found no link between low back pain and LSTV….
Cureus. 2022 Jun 26;14(6):e26341. doi: 10.7759/cureus.26341. eCollection 2022 Jun.
Introduction Bertolotti's syndrome (BS) describes the relationship between low back pain (LBP) and lumbosacral transitional vertebra (LSTV). It is a factor that is sometimes overlooked when it comes to evaluating and treating LBP.Because of the different diagnostic modalities and criteria used in the research, the LSTV incidence in the general population varies greatly, and hence the link between LSTV and LBP remains contentious. Some researchers found no link between low back pain and LSTV. As a result, the management of BS remains controversial and multiple treatments have been suggested, including locally injected steroid and various surgical approaches. Methods This retrospective cohort study included a total of 288 patients who underwent lumbosacral surgical procedures for disc prolapse, lumbar canal stenosis, spondylolithesis and post-laminectomy syndrome during the period between January 2016 and May 2020. Trauma, tumours and scoliotic patients were excluded. All data were collected from the departmental database. All cases were done by the same surgical team at El Hadra University Hospital Spine Unit, Egypt. The patients were divided into two groups. Group A consisted of 133 patients in whom LSTV was detected by radiologic findings. In contrast, Group B consisted of 155 patients in whom LSTV was not detected. Results In our study, the overall prevalence of LSTV among 288 patients who underwent lumbosacral surgical interventions was 46.2%. On comparing the incidence of surgical interventions between both groups, there was non-significant difference in most of surgical interventions. The incidence of L3-5 double-level posterior lumbar interbody fusion (PLIF) among LSTV patients was 16.5% compared to 4.61% in the other group. The incidence of L4-S1 double-level PLIF among LSTV patients was 15.04% compared to 7.24% in the other group. Regarding adjacent segment pathology, the incidence of lumbar canal stenosis and degenerative spondylolithesis was higher in the LSTV group (20.3% and 11.3%, respectively) compared to the non-LSTV group (9.7% and 5.2%, respectively). The incidence of disc prolapse was lower in the LSTV group (56.39%) compared to the non-LSTV group (71.0%). There was a non-significant difference between the incidence of lytic spondylolithesis and postlaminectomy syndrome between both groups. Conclusion The overall prevalence of LSTV among all cases who underwent lumbosacral surgical procedures at the El Hadra University Hospital was 46.2%. The incidence of lumbar canal stenosis and degenerative spondylolithesis was higher in the LSTV group compared to the non-LSTV group. However, the incidence of disc prolapse was lower in the LSTV group compared to the non-LSTV group. The incidence of disc prolapse and degenerative spondylolithesis at the L4-5 level was higher in the LSTV group compared to the non-LSTV group. In contrast, the incidence at L5-S1 was lower in the LSTV group compared to the non-LSTV group. Hence, LSTV is considered a risk factor for disc degenerative changes at the level above the transitional vertebra level.
PMID:35903576 | PMC:PMC9318490 | DOI:10.7759/cureus.26341
Prevalence of Bertolotti's Syndrome in Lumbosacral Surgery Procedures
Lumbar Decompression for Back Pain
Lumbar Decompression for Back Pain | Piriformis syndrome symptoms uk
Lumbar Decompression Surgery | Maverick disc
Cureus. 2022 Jun 26;14(6):e26341. doi: 10.7759/cureus.26341. eCollection 2022 Jun.ABSTRACTIntroduction Bertolotti's syndrome (BS) describes the relationship between low back pain (LBP) and lumbosacral transitional vertebra (LSTV). It is a factor that is sometimes overlooked when it comes to evaluating and treating LBP.Because of the different diagnostic modalities and criteria used in the research, the LSTV incidence…