Fix LASSO (first two of #342) (#343)

* Fix LASSO (#342)
* change loss function in doc to match code
* allow `n == p` case
* lasso add test_full_rank_x

---------

Co-authored-by: Zhou Xiaozhou <zxz@jiweifund.com>

src/linear/lasso.rs

@@ -9,7 +9,7 @@
 //!
 //! Lasso coefficient estimates solve the problem:
 //!
-//! \\[\underset{\beta}{minimize} \space \space \sum_{i=1}^n \left( y_i - \beta_0 - \sum_{j=1}^p \beta_jx_{ij} \right)^2 + \alpha \sum_{j=1}^p \lVert \beta_j \rVert_1\\]
+//! \\[\underset{\beta}{minimize} \space \space \frac{1}{n} \sum_{i=1}^n \left( y_i - \beta_0 - \sum_{j=1}^p \beta_jx_{ij} \right)^2 + \alpha \sum_{j=1}^p \lVert \beta_j \rVert_1\\]
 //!
 //! This problem is solved with an interior-point method that is comparable to coordinate descent in solving large problems with modest accuracy,
 //! but is able to solve them with high accuracy with relatively small additional computational cost.
@@ -246,7 +246,7 @@ impl<TX: FloatNumber + RealNumber, TY: Number, X: Array2<TX>, Y: Array1<TY>> Las
     pub fn fit(x: &X, y: &Y, parameters: LassoParameters) -> Result<Lasso<TX, TY, X, Y>, Failed> {
         let (n, p) = x.shape();
 
-        if n <= p {
+        if n < p {
             return Err(Failed::fit(
                 "Number of rows in X should be >= number of columns in X",
             ));
@@ -369,6 +369,7 @@ impl<TX: FloatNumber + RealNumber, TY: Number, X: Array2<TX>, Y: Array1<TY>> Las
 #[cfg(test)]
 mod tests {
     use super::*;
+    use crate::linalg::basic::arrays::Array;
     use crate::linalg::basic::matrix::DenseMatrix;
     use crate::metrics::mean_absolute_error;
 
@@ -448,6 +449,36 @@ mod tests {
         assert!(mean_absolute_error(&y_hat, &y) < 2.0);
     }
 
+    #[cfg_attr(
+        all(target_arch = "wasm32", not(target_os = "wasi")),
+        wasm_bindgen_test::wasm_bindgen_test
+    )]
+    #[test]
+    fn test_full_rank_x() {
+        // x: randn(3,3) * 10, demean, then round to 2 decimal points
+        // y = x @ [10.0, 0.2, -3.0], round to 2 decimal points
+        let param = LassoParameters::default()
+            .with_normalize(false)
+            .with_alpha(200.0);
+        let x = DenseMatrix::from_2d_array(&[
+            &[-8.9, -2.24, 8.89],
+            &[-4.02, 8.89, 12.33],
+            &[12.92, -6.65, -21.22],
+        ])
+        .unwrap();
+
+        let y = vec![-116.12, -75.41, 191.53];
+        let w = Lasso::fit(&x, &y, param)
+            .unwrap()
+            .coefficients()
+            .iterator(0)
+            .copied()
+            .collect();
+
+        let expected_w = vec![5.20289531, 0., -5.32823882]; // by coordinate descent
+        assert!(mean_absolute_error(&w, &expected_w) < 1e-3); // actual mean_absolute_error is about 2e-4
+    }
+
     // TODO: serialization for the new DenseMatrix needs to be implemented
     // #[cfg_attr(all(target_arch = "wasm32", not(target_os = "wasi")), wasm_bindgen_test::wasm_bindgen_test)]
     // #[test]