Questions	Answers
We got several questions about how many parameters, features for Lasso regression you can use relatively to number of data points	The number of features you can include in a Lasso regression relative to the number of data points depends on various factors, including the structure and quality of the data, the degree of multicollinearity among the predictors, and the actual relationships between predictors and the response variable.
However, a common rule of thumb for traditional linear regression models is that you should have at least 10-20 data points for each predictor. This guideline can be relaxed for Lasso regression due to its ability to perform feature selection and handle multicollinearity.
Lasso regression shrinks the coefficients of irrelevant features towards zero, effectively performing feature selection. This allows Lasso to handle datasets with a large number of features, even when the number of observations is relatively low.
Here are some general considerations:

1. Sparsity: If the true model is sparse (i.e., only a few predictors truly influence the response), then Lasso can handle a larger number of predictors relative to the sample size.
2. Multicollinearity: Lasso handles multicollinearity well by selecting one predictor from a group of correlated predictors and shrinking the others towards zero.
3. Sample Size: Even though Lasso can handle many predictors, having an extremely low ratio of observations to predictors can still pose risks, such as overfitting. Regularization strength (controlled by the hyperparameter lambda is crucial in such cases.
4. Cross-Validation: It's essential to use cross-validation to select the optimal regularization parameter lambda and to assess model performance.
5. Additional Data: If model performance isn't satisfactory and you suspect overfitting, one option is to gather more data, if feasible. Remember, while Lasso provides flexibility in handling many features, the interpretability, generalizability, and predictive power of the model are paramount. Always check the model's performance on unseen data (e.g., using a validation set or cross-validation) to ensure that it generalizes well. | | Why tsne in R library first run PCA | The Rtsne function in the R package for t-SNE (t-distributed Stochastic Neighbor Embedding) can employ PCA (Principal Component Analysis) as an optional preprocessing step. Here's why:
6. Speed: t-SNE has a computational complexity of O(n^2) (where n is the number of data points) for its pairwise distance calculations, making it quite slow for large datasets. Running PCA before t-SNE reduces the dimensionality of the data and therefore speeds up t-SNE.
7. Noise Reduction: PCA, being a linear dimensionality reduction technique, captures the major variance in data. By projecting data to its principal components, you can filter out some of the noise, potentially leading to clearer t-SNE visualizations.
8. Memory Efficiency: By reducing dimensions using PCA, t-SNE will consume less memory, which can be crucial when dealing with large datasets.
9. Initial Embedding: t-SNE's performance (both speed and quality of results) can sometimes be improved by providing a good initial solution. The results from PCA can be used as this initial solution. In the Rtsne function, the PCA step is controlled by the pca parameter. If you set pca = TRUE (which is the default), it will first reduce the data to 50 principal components using PCA before applying t-SNE. If you believe that PCA might discard important nonlinear relationships in your data, you can set pca = FALSE. However, keep in mind that this might make the t-SNE process slower, especially for high-dimensional data. |