diff --git a/doc/modules/decomposition.rst b/doc/modules/decomposition.rst
index 54ed20cc36be4..e58796fa85b46 100644
--- a/doc/modules/decomposition.rst
+++ b/doc/modules/decomposition.rst
@@ -231,7 +231,7 @@ problem solved is a PCA problem (dictionary learning) with an
 .. math::
    (U^*, V^*) = \underset{U, V}{\operatorname{arg\,min\,}} & \frac{1}{2}
                 ||X-UV||_{\text{Fro}}^2+\alpha||V||_{1,1} \\
-                \text{subject to } & ||U_k||_2 = 1 \text{ for all }
+                \text{subject to } & ||U_k||_2 <= 1 \text{ for all }
                 0 \leq k < n_{components}
 
 :math:`||.||_{\text{Fro}}` stands for the Frobenius norm and :math:`||.||_{1,1}`
@@ -513,7 +513,7 @@ dictionary fixed, and then updating the dictionary to best fit the sparse code.
 .. math::
    (U^*, V^*) = \underset{U, V}{\operatorname{arg\,min\,}} & \frac{1}{2}
                 ||X-UV||_{\text{Fro}}^2+\alpha||U||_{1,1} \\
-                \text{subject to } & ||V_k||_2 = 1 \text{ for all }
+                \text{subject to } & ||V_k||_2 <= 1 \text{ for all }
                 0 \leq k < n_{\mathrm{atoms}}
 
 
diff --git a/doc/whats_new/v1.0.rst b/doc/whats_new/v1.0.rst
index 17c2b845d40d9..1cc84b74198ce 100644
--- a/doc/whats_new/v1.0.rst
+++ b/doc/whats_new/v1.0.rst
@@ -2,6 +2,26 @@
 
 .. currentmodule:: sklearn
 
+.. _changes_1_0_2:
+
+Version 1.0.2
+=============
+
+**In Development**
+
+Changelog
+---------
+
+:mod:`sklearn.decomposition`
+............................
+
+- |Fix| Fixed the constraint on the objective function of
+  :class:`decomposition.DictionaryLearning`,
+  :class:`decomposition.MiniBatchDictionaryLearning`, :class:`decomposition.SparsePCA`
+  and :class:`decomposition.MiniBatchSparsePCA` to be convex and match the referenced
+  article. :pr:`19210` by :user:`Jérémie du Boisberranger <jeremiedbb>`.
+
+
 .. _changes_1_0_1:
 
 Version 1.0.1
diff --git a/sklearn/decomposition/_dict_learning.py b/sklearn/decomposition/_dict_learning.py
index e4edaf31c9c32..aee341bd88b05 100644
--- a/sklearn/decomposition/_dict_learning.py
+++ b/sklearn/decomposition/_dict_learning.py
@@ -480,8 +480,8 @@ def _update_dict(
         if positive:
             np.clip(dictionary[k], 0, None, out=dictionary[k])
 
-        # Projection on the constraint set ||V_k|| == 1
-        dictionary[k] /= linalg.norm(dictionary[k])
+        # Projection on the constraint set ||V_k|| <= 1
+        dictionary[k] /= max(linalg.norm(dictionary[k]), 1)
 
     if verbose and n_unused > 0:
         print(f"{n_unused} unused atoms resampled.")
@@ -1331,7 +1331,7 @@ class DictionaryLearning(_BaseSparseCoding, BaseEstimator):
 
         (U^*,V^*) = argmin 0.5 || X - U V ||_Fro^2 + alpha * || U ||_1,1
                     (U,V)
-                    with || V_k ||_2 = 1 for all  0 <= k < n_components
+                    with || V_k ||_2 <= 1 for all  0 <= k < n_components
 
     ||.||_Fro stands for the Frobenius norm and ||.||_1,1 stands for
     the entry-wise matrix norm which is the sum of the absolute values
@@ -1608,7 +1608,7 @@ class MiniBatchDictionaryLearning(_BaseSparseCoding, BaseEstimator):
 
        (U^*,V^*) = argmin 0.5 || X - U V ||_Fro^2 + alpha * || U ||_1,1
                     (U,V)
-                    with || V_k ||_2 = 1 for all  0 <= k < n_components
+                    with || V_k ||_2 <= 1 for all  0 <= k < n_components
 
     ||.||_Fro stands for the Frobenius norm and ||.||_1,1 stands for
     the entry-wise matrix norm which is the sum of the absolute values