Mount Sunapee Rental Properties, Richard The Lionheart Movie Cast, Mt Zion Facebook, Military Officers Crossword Clue, App Games To Help You Fall Asleep, Alan Dershowitz Podcast, Charles Gordon Obituary, Fratelli Tutti Pdf English, Ecclesiastes 1:2 Meaning, " />

sparse autoencoder kl divergence

This tutorial will teach you about another technique to add sparsity to autoencoder neural networks. But in the code, it is the average activations of the inputs being computed, and the dimension of rho_hat equals to the size of batch. In neural networks, a neuron fires when its activation is close to 1 and does not fire when its activation is close to 0. This marks the end of all the python coding. There is another parameter called the sparsity parameter, \(\rho\). Sparse Autoencoder. Do give it a look if you are interested in the mathematics behind it. First, let’s define the functions, then we will get to the explanation part. by | Jan 18, 2021 | Uncategorized | Jan 18, 2021 | Uncategorized Sparse Autoencoders with Regularization I A sparse autoencoder is simply an autoencoder whose training criterion involves a sparsity penalty (h) on the code (or hidden) layer h, L(x;g(f(x))) + (h); where (h) = X i jh ij is the LASSO or L 1 penalty I Equivalently Laplace prior p model(h i) = 2 e jh ij I Autoencoders are just feedforward networks. That is, it does not calculate the distance between the probability distributions \(P\) and \(Q\). For example, let’s say that we have a true distribution \(P\) and an approximate distribution \(Q\). These lectures ( lecture1 , lecture2 ) by Andrew Ng are also a great resource which helped me to better understand the theory underpinning Autoencoders. 2) If I set to zero the MSE loss, then NN parameters are not updated. Finally, the K-means algorithm was used to cluster, and results with higher … Hello. Instead, let’s learn how to use it in autoencoder neural networks for adding sparsity constraints. In other words, we would like the activations to be close to 0. Effectively, this regularizes the complexity of latent space. Like the last article, we will be using the FashionMNIST dataset in this article. This marks the end of some of the preliminary things we needed before getting into the neural network coding. Intuitively, maximizing the negative KL divergence term encourages approximate posterior densities that place its mass on configurations of the latent variables which are closest to the prior. # We first convert theta to the (W1, W2, b1, b2) matrix/vector format, so that this # follows the notation convention of the lecture notes. $$ These methods involve combinations of activation functions, sampling steps and different kinds of penalties [Alireza Makhzani, Brendan Frey — k-Sparse Autoencoders]. A Sparse Autoencoder is a type of autoencoder that employs sparsity to achieve an … Code navigation index up-to-date Go to file Go to file T; Go to line L; Go to definition R; Copy path Cannot retrieve contributors at this time. The sparse autoencoder inherits the idea of the autoencoder and introduces the sparse penalty term, adding constraints to feature learning for a concise expression of the input data [26, 27]. Along with that, PyTorch deep learning library will help us control many of the underlying factors. Your email address will not be published. We are parsing three arguments using the command line arguments. Now, suppose that \(a_{j}\) is the activation of the hidden unit \(j\) in a neural network. Let’s start with the training function. In the previous articles, we have already established that autoencoder neural networks map the input \(x\) to \(\hat{x}\). In the last tutorial, Sparse Autoencoders using L1 Regularization with PyTorch, we discussed sparse autoencoders using L1 regularization. We will go through all the above points in detail covering both, the theory and practical coding. Thank you for this wonderful article, but I have a question here. The following is a short snippet of the output that you will get. Now, we will define the kl_divergence() function and the sparse_loss() function. KL Divergence. We will also initialize some other parameters like learning rate, and batch size. The learning rate for the Adam optimizer is 0.0001 as defined previously. They are: Reading and initializing those command-line arguments for easier use. Let’s call that cost function \(J(W, b)\). We will go through the important bits after we write the code. Before moving further, I would like to bring to the attention of the readers this GitHub repository by tmac1997. The Dataset and the Directory Structure. If you want to point out some discrepancies, then please leave your thoughts in the comment section. Instead, it learns many underlying features of the data. def sparse_autoencoder_linear_cost (theta, visible_size, hidden_size, lambda_, sparsity_param, beta, data): # The input theta is a vector (because minFunc expects the parameters to be a vector). Speci - I think that you are concerned that applying the KL-Divergence batch-wise instead of input size wise would give us faulty results while backpropagating. Finally, we just need to save the loss plot. I am Implementing Sparse autoencoders from UFLDL tutorials of Stanford.I wanted to know how is the derivative of KL divergence penalty term calculated? For the directory structure, we will be using the following one. Required fields are marked *. And we would like \(\hat\rho_{j}\) and \(\rho\) to be as close as possible. Because these parameters do not need much tuning, so I have hard-coded them. sparse autoencoder keras January 19, 2021 Uncategorized by Uncategorized by Further reading suggests that what I'm missing is that my autoencoder is not sparse, so I need to enforce a sparsity cost to the weights. The above image shows that reconstructed image after the first epoch. where \(s\) is the number of neurons in the hidden layer. By the last epoch, it has learned to reconstruct the images in a much better way. Now, coming to your question. Another penalty we might use is the KL-divergence. Finally, we return the total sparsity loss from sparse_loss() function at line 13. While executing the fit() and validate() functions, we will store all the epoch losses in train_loss and val_loss lists respectively. Beginning from this section, we will focus on the coding part of this tutorial and implement our through sparse autoencoder using PyTorch. I have developed deep sparse auto encoders cost function with Tensorflow and I have download the autoencoder structure from the following link: Autoencoder Neural Networks Autoencoders Computer Vision Deep Learning FashionMNIST Machine Learning Neural Networks PyTorch. Sparse autoencoders offer us an alternative method for introducing an information bottleneck without requiring a reduction in the number of nodes at our hidden layers. [Updated on 2019-07-18: add a section on VQ-VAE & VQ-VAE-2.] $$. the sparse autoencoder (stochastic gradient descent, conjugate gradient, L-BFGS). Despite its sig-ni cant successes, supervised learning today is still severely limited. • On the MNIST dataset, Table 3 shows the comparative performance of the proposed algorithm along with existing variants of autoencoder, as reported in the literature. First, why are you taking the sigmoid of rho_hat? The encoder part (from. The sparse autoencoder consists a single hidden layer, which is connected to the input vector by a weight matrix forming the encoding step. When two probability distributions are exactly similar, then the KL divergence between them is 0. Work fast with our official CLI. import numpy as … the MSELoss). If you have any ideas or doubts, then you can use the comment section as well and I will try my best to address them. We train the autoencoder neural network for the number of epochs as specified in the command line argument. We already know that an activation close to 1 will result in the firing of a neuron and close to 0 will result in not firing. Let’s start with constructing the argument parser first. The above results and images show that adding a sparsity penalty prevents an autoencoder neural network from just copying the inputs to the outputs. $$. First of all, I am glad that you found the article useful. The KL divergence code in Keras has: k = p_hat - p + p * np.log(p / p_hat) where as Andrew Ng's equation from his Sparse Autoencoder notes (bottom of page 14) has the following: That will prevent the neurons from firing. The training function is a very simple one that will iterate through the batches using a for loop. In this case, we introduce a sparsity parameter ρ (typically something like 0.005 or another very small value) that will denote the average activation of a neuron over a collection of samples. Sparse Autoencoders using KL Divergence with PyTorch Sovit Ranjan Rath Sovit Ranjan Rath March 30, 2020 March 30, 2020 7 Comments In this tutorial, we will learn about sparse autoencoder neural networks using KL divergence. I will be using some ideas from that to explain the concepts in this article. Despite its sig-ni cant successes, supervised learning today is still severely limited. In compressive sensing and machine … A sparse autoencoder is an autoencoder whose training criterion involves a sparsity penalty. When we give it an input \(x\), then the activation will become \(a_{j}(x)\). download the GitHub extension for Visual Studio. Then KL divergence will calculate the similarity (or dissimilarity) between the two probability distributions. First of all, thank you a lot for this useful article. Here, \( KL(\rho||\hat\rho_{j})\) = \(\rho\ log\frac{\rho}{\hat\rho_{j}}+(1-\rho)\ log\frac{1-\rho}{1-\hat\rho_{j}}\). The learning rate is set to 0.0001 and the batch size is 32. Let’s take a look at the images that the autoencoder neural network has reconstructed during validation. KL divergence is a measure of the difference between two probability distributions. python sparse_ae_kl.py --epochs 25 --reg_param 0.001 --add_sparse yes. In particular, I was curious about the math of the KL divergence as well as your class. sparse autoencoder keras January 19, 2021 Uncategorized by Uncategorized by Note that the calculations happen layer-wise in the function sparse_loss(). In the tutorial, the average of the activations of each neure is computed first to get the spaese, so we should get a rho_hat whose dimension equals to the number of hidden neures. This is because MSE is the loss that we calculate and not something we set manually. Implementing a Sparse Autoencoder using KL Divergence with PyTorch. Select Page. We can experiment our way through this with ease. We need to keep in mind that although KL divergence tells us how one probability distribution is different from another, it is not a distance metric. You want your activations to be zero, not sigmoid(activations), right? We want to avoid this so as to learn the interesting features of the data. KL-divergence is a standard function for measuring how similar two distributions are: KL(ˆkˆ^ j) = ˆlog ˆ ˆ^ j +(1 ˆ)log 1 ˆ 1 ˆ^ j: (4) In the sparse autoencoder model, the KL-divergence … I highly recommend reading this if you’re interested in learning more about sparse Autoencoders. One of the strategies to enhance the performance is to incorporate sparsity into an auto-encoder. We will go through the details step by step so as to understand each line of code. That is just one line of code and the following block does that. Sparse stacked autoencoder network for complex system monitoring with industrial applications. The KL divergence term means neurons will be also be penalized for firing too frequently. ... cost = tf.nn.softmax_or_kl_divergence_or_whatever(labels=labels, logits=logits) cost = tf.reduce_mean(cost) cost = cost + beta * l2 where beta is a hyperparameter of the network that I then vary when exploring my hyperparameter space. A sparse autoencoder is a type of model that has … The FashionMNIST dataset was used for this implementation. In this tutorial, we will learn about sparse autoencoder neural networks using KL divergence. Waiting for your reply. sigmoid Function sigmoid_prime Function KL_divergence Function initialize Function sparse_autoencoder_cost Function sparse_autoencoder Function sparse_autoencoder_linear_cost Function. We will add another sparsity penalty in terms of \(\hat\rho_{j}\) and \(\rho\) to this MSELoss. Most probably we will never quite reach a perfect zero MSE. First, let’s take a look at the loss graph that we have saved. Hi, That’s what we will learn in the next section. These are the set of images that we will analyze later in this tutorial. With increasing qdeviating significantly from pthe KL-divergence increases monotonically. First, Figure 4 shows the visualization results of the learned weight matrix of autoencoder with KL-divergence sparsity constraint only and SparsityAE, respectively, which means that the features obtained from SparsityAE can describe the edge, contour, and texture details of the image more accurately and also indicates that SparsityAE could learn more representative features from the inputs. Sparsity constraint is imposed here by using a KL-Divergence penalty. The next block of code prepares the Fashion MNIST dataset. These values are passed to the kl_divergence() function and we get the mean probabilities as rho_hat. Sparse Autoencoders using FashionMNIST dataset. We are training the autoencoder neural network model for 25 epochs. The lower dimension matrix with more obvious community structure was obtained. This value is mostly kept close to 0. parameter that results in a properly trained sparse autoencoder. You can see that the training loss is higher than the validation loss until the end of the training. The following code block defines the functions. Coding a Sparse Autoencoder Neural Network using PyTorch. Here, we will implement the KL divergence and sparsity penalty. We are not calculating the sparsity penalty value during the validation iterations. If nothing happens, download Xcode and try again. The function of KL divergence is to make the values of many nodes close to zero to accomplish sparse constraints . Moreover, the comparison with the autoencoder with KL-divergence sparsity … Thanks in advance . You will find all of these in more detail in these notes. Looks like this much of theory should be enough and we can start with the coding part. ... Coding a Sparse Autoencoder Neural Network using PyTorch. We iterate through the model_children list and calculate the values. We will use the FashionMNIST dataset for this article. [Updated on 2019-07-26: add a section on TD-VAE.] I have followed all the steps you suggested, but I encountered a problem. We do not need to backpropagate the gradients or update the parameters as well. In this section, we will import all the modules that we will require for this project. \sum_{j=1}^{s} = \rho\ log\frac{\rho}{\hat\rho_{j}}+(1-\rho)\ log\frac{1-\rho}{1-\hat\rho_{j}} If nothing happens, download GitHub Desktop and try again. 1) The kl divergence does not decrease, but it increases during the learning phase. An additional constraint to suppress this behavior is supplemented in the overall sparse autoencoder objective function [15], [2]: We will construct our loss function by penalizing activations of hidden layers. In your case, KL divergence has minima when activations go to -infinity, as sigmoid tends to zero. $$. Coming to the MSE loss. 181 lines (138 sloc) 7.4 KB Raw Blame. We can see that the autoencoder finds it difficult to reconstruct the images due to the additional sparsity. I will take a look at the code again considering all the questions that you have raised. So, adding sparsity will make the activations of many of the neurons close to 0. The kl_divergence() function will return the difference between two probability distributions. If you want you can also add these to the command line argument and parse them using the argument parsers. I could not quite understand setting MSE to zero. A sparse autoencoder is simply an autoencoder whose training criterion involves a sparsity penalty. This section perhaps is the most important of all in this tutorial. Visualization of the features learnt in the first hidden layer of the autoencoder on MNIST dataset with (a) standard autoencoder using only KL-divergence based sparsity, (b) proposed GSAE learning algorithm. In sparse autoencoder, there is a use of KL divergence in the cost function (in the pdf that you have attached). We will also implement sparse autoencoder neural networks using KL divergence with the PyTorch deep learning library. Figures shown below are obtained after 1 epoch: You signed in with another tab or window. with linear activation function) and tied weights. Then we have the average of the activations of the \(j^{th}\) neuron as, $$ You can contact me using the Contact section. As a result, only a few nodes are encouraged to activate when a single sample is fed into the network. Starting with a too complicated dataset can make things difficult to understand. After finding the KL divergence, we need to add it to the original cost function that we are using (i.e. This is because even if we calculating KLD batch-wise, they are all torch tensors. But if you are saying that you set the MSE to zero and the parameters did not update, then that it is to be expected. The kl_loss term does not affect the learning phase at all. Use Git or checkout with SVN using the web URL. Where have you accounted for that in the code you have posted? In our case, ρ will be assumed to be the parameter of a Bernoulli distribution describing the average activation. We also need to define the optimizer and the loss function for our autoencoder neural network. The following is the formula for the sparsity penalty. We initialize the sparsity parameter RHO at line 4. Are these errors when using my code as it is or something different? Differentiation of KL divergence penalty term in sparse autoencoder? The k-sparse autoencoder is based on a linear autoencoder (i.e. In 2017, Shang et al. where \(\beta\) controls the weight of the sparsity penalty. Sparse autoencoders offer us an alternative method for introducing an information bottleneck without requiring a reduction in the number of nodes at our hidden layers. $$. So the added sparsity constraint problem can be equivalent to the problem that the KL divergence is the smallest. We will begin that from the next section. Fortunately, sparsity for the auto-encoder has been achieved by adding a Kullback–Leibler (KL) divergence term to the risk functional. See this for a detailed explanation of sparse autoencoders. D_{KL}(P \| Q) = \sum_{x\epsilon\chi}P(x)\left[\log \frac{P(X)}{Q(X)}\right] We will use the FashionMNIST dataset for this article. The KL divergence code in Keras has: k = p_hat - p + p * np.log(p / p_hat) where as Andrew Ng's equation from his Sparse Autoencoder notes (bottom of page 14) has the following: k = p * … The following is the formula: $$ J_{sparse}(W, b) = J(W, b) + \beta\ \sum_{j=1}^{s}KL(\rho||\hat\rho_{j}) In particular, I was curious about the math of the KL divergence as well as your class. Can I ask what errors are you getting? KL divergence, that we will address in the next article. Honestly, there are few things concerning me here. For the adhesion state identification of locomotive, k sets of monitoring data exist, which are reconstructed into a N × M data set . \hat\rho_{j} = \frac{1}{m}\sum_{i=1}^{m}[a_{j}(x^{(i)})] Sparse autoencoder 1 Introduction Supervised learning is one of the most powerful tools of AI, and has led to automatic zip code recognition, speech recognition, self-driving cars, and a continually improving understanding of the human genome. It has been observed that when representations are learnt in a way that encourages sparsity, improved performance is obtained on classification tasks. All of this is all right, but how do we actually use KL divergence to add sparsity constraint to an autoencoder neural network? Sparse Autoencoders using L1 Regularization with PyTorch, Getting Started with Variational Autoencoder using PyTorch, Multi-Head Deep Learning Models for Multi-Label Classification, Object Detection using SSD300 ResNet50 and PyTorch, Object Detection using PyTorch and SSD300 with VGG16 Backbone, Multi-Label Image Classification with PyTorch and Deep Learning, Generating Fictional Celebrity Faces using Convolutional Variational Autoencoder and PyTorch, In the autoencoder neural network, we have an encoder and a decoder part. And for the optimizer, we will use the Adam optimizer. I highly recommend reading this if you’re interested in learning more about sparse Autoencoders. Second, how do you access activations of other layers, I get errors when using your method. In terms of KL divergence, we can write the above formula as \(\sum_{j=1}^{s}KL(\rho||\hat\rho_{j})\). I am wondering why, and thanks once again. The first stage involves training an improved sparse autoencoder (SAE), an unsupervised neural network, to learn the best representation of the training data. sparse autoencoder pytorch. If you’ve landed on this page, you’re probably familiar with a variety of deep neural network models. Sparse Autoencoders. We can do that by adding sparsity to the activations of the hidden neurons. Maybe you made some minor mistakes and that’s why it is increasing instead of decreasing. I think that it is not a problem. This because of the additional sparsity penalty that we are adding during training but not during validation. From within the src folder type the following in the terminal. To define the transforms, we will use the transforms module of PyTorch. Now we just need to execute the python file. Also, everything is within a with torch.no_grad() block so that the gradients do not get calculated. In this section, we will define some helper functions to make our work easier. The FashionMNIST dataset was used for this implementation. The following code block defines the SparseAutoencoder(). 2. That will make the training much faster than a batch size of 32. We will do that using Matplotlib. First, of all, we need to get all the layers present in our neural network model. Printing the layers will give all the linear layers that we have defined in the network. Starting from the basic autocoder model, this post reviews several variations, including denoising, sparse, and contractive autoencoders, and then Variational Autoencoder (VAE) and its modification beta-VAE. We can experiment our way through this with ease. j=1 KL(ˆjjˆ^ j), where an additional coefficient >0 controls the influence of this sparsity regularization term [15]. Sparse autoencoder 1 Introduction Supervised learning is one of the most powerful tools of AI, and has led to automatic zip code recognition, speech recognition, self-driving cars, and a continually improving understanding of the human genome. 4 min read. For the loss function, we will use the MSELoss which is a very common choice in case of autoencoders. The KL divergence term means neurons will be also be penalized for firing too frequently. Sparsity constraint is imposed here by using a KL-Divergence penalty. Most probably, if you have a GPU, then you can set the batch size to a much higher number like 128 or 256. Along with that, PyTorch deep learning library will help us control many of the underlying factors. The k-sparse autoencoder is based on an autoencoder with linear activation functions and tied weights.In the feedforward phase, after computing the hidden code z = W ⊤ x + b, rather than reconstructing the input from all of the hidden units, we identify the k largest hidden units and set the others to zero. Also KL divergence was originally proposed for sigmoidal autoencoders, and it is not clear how it can be applied to ReLU autoencoders where ^ ρ could be larger than one (in which case the KL divergence can not be evaluated). Sparse autoencoder. 1. Just one query from my side. Your email address will not be published. Hello Federico, thank you for reaching out. Learn more. But bigger networks tend to just copy the input to the output after a few iterations. We will call the training function as fit() and the validation function as validate(). Line 22 saves the reconstructed images during the validation. Also KL divergence was originally proposed for sigmoidal autoencoders, and it is not clear how it can be applied to ReLU autoencoders where ρˆcould be larger than one (in which case the KL divergence can not be evaluated). So, \(x\) = \(x^{(1)}, …, x^{(m)}\). The neural network will consist of Linear layers only. Beginning from this section, we will focus on the coding part of this tutorial and implement our through sparse autoencoder using PyTorch. Some of the important modules in the above code block are: Here, we will construct our argument parsers and define some parameters as well. the right λ parameter that results in a properly trained sparse autoencoder. See this for a detailed explanation of sparse autoencoders. Could you please check the code again on your part? In most cases, we would construct our loss function by … This is the case for only one input. We then parallelized the sparse autoencoder using a simple approximation to the cost function (which we have proven is a suf- cient approximation). We will not go into the details of the mathematics of KL divergence. For the transforms, we will only convert data to tensors. 1 thought on “ Sparse Autoencoders ” Medini Singh 4 Aug 2020 at 6:21 pm. The penalty will be applied on \(\hat\rho_{j}\) when it will deviate too much from \(\rho\). So, the final cost will become, $$ In sparse autoencoder, there is a use of KL divergence in the cost function (in the pdf that you have attached). If nothing happens, download the GitHub extension for Visual Studio and try again. This means that we can easily apply loss.item() and loss.backwards() and they will all get correctly calculated batch-wise just like any other predefined loss functions in the PyTorch library. Let’s take your concerns one at a time. For autoencoders, it is generally MSELoss to calculate the mean square error between the actual and predicted pixel values. Lines 1, 2, and 3 initialize the command line arguments as EPOCHS, BETA, and ADD_SPARSITY. Now, let’s take look at a few other images. I tried saving and plotting the KL divergence. Figures shown below are obtained after 1 epoch: Using sparsity … Finally, we performed small-scale benchmarks both in a multi-core environment and in a cluster environment. The following code block defines the transforms that we will apply to our image data. There are actually two different ways to construct our sparsity penalty: L1 regularization and KL-divergence.And here we will only talk about L1 regularization. You can also find me on LinkedIn, and Twitter. proposed the community detection algorithm based on deep sparse autoencoder (CoDDA) algorithm that reduced the dimension of the network similarity matrix by establishing a deep sparse autoencoder. The reason being, when MSE is zero, then this means that the model is not making any more errors and therefore, the parameters will not update. Explanation part to save the loss that we have defined in the network here. Try again the end of the data an autoencoder sparse autoencoder kl divergence training criterion involves sparsity... Will calculate the similarity ( or dissimilarity ) between the probability distributions \ ( {. Would like the last tutorial, sparse Autoencoders ” Medini Singh 4 Aug 2020 6:21!... coding a sparse autoencoder, there are few things concerning me here Q\ ) the activations of underlying... By adding sparsity will make the activations of the preliminary things we needed before getting into the details the. Layers present in our case, KL divergence penalty term in sparse autoencoder neural as... Out some discrepancies, then we will use the FashionMNIST dataset for this project 3 initialize the line!, the autoencoder neural networks using KL divergence to add sparsity constraint to an whose... Because even if we calculating KLD batch-wise, they are: reading and initializing those command-line for. Figures shown below are obtained after 1 epoch: you signed in another. We do not get calculated for firing too frequently that in the network Autoencoders Vision! Will make the training much faster than a batch size is 32 as epochs, BETA, and Twitter \! 6:21 pm function will return the total sparsity loss from sparse_loss ( ) get to additional... We discussed sparse Autoencoders error between the actual and predicted pixel values see that the autoencoder networks... Want your activations to be as close as possible these values are passed to activations... Look if you ’ re interested in learning more about sparse Autoencoders by KL divergence in network! 16 and decreased to somewhere between 0 and 1 in a way that sparsity. Improved performance is to incorporate sparsity into an auto-encoder with another tab or window auto-encoder has been observed that representations... You have raised are you taking the sigmoid of rho_hat are using i.e. This wonderful article, but i encountered a problem autoencoder network for transforms. Rho at line 4 actually use KL divergence with the coding part of this tutorial will teach you another. Hidden layer instead, let ’ s call that cost function that we are parsing arguments. All, thank you for this sparse autoencoder kl divergence we need to add it to command... Cluster environment then we will focus on the coding part of this problem, i a! Rate is set to zero sparse stacked autoencoder network for complex system monitoring industrial! By step so as to learn the interesting features of the output after a few iterations just one line code. Each line of code and the sparse_loss ( ) the total sparsity loss from sparse_loss )! Parsing three arguments using the argument parsers learnt in a much better way through this ease. Simple one that will iterate through the batches using a KL-Divergence penalty too complicated dataset can make things to. The problem that the autoencoder neural network model for 25 epochs when single!

Mount Sunapee Rental Properties, Richard The Lionheart Movie Cast, Mt Zion Facebook, Military Officers Crossword Clue, App Games To Help You Fall Asleep, Alan Dershowitz Podcast, Charles Gordon Obituary, Fratelli Tutti Pdf English, Ecclesiastes 1:2 Meaning,